I discovered a torn string by the bottom of my Tzitzis, by the area that rests on the corner, in middle of Davening. What do I do?

The Tallis is Pasul, and hence must be removed as soon as possible. If you have a Kosher replacement Tallis, then put it on, and you are to also say a new blessing over it, if you are in an area of prayer in which it is permitted to do so.

Detailed answer with Background of laws:

Is the Tallis Passul?^[1] Yes. Background: If even one string tore from the root of the fringes, the Tzitzis is invalid.^[2] This means that if even one string tore near the corner hole, within the area that precedes the braid^[3], or within any area of the braid^[4], the Tzitzis is invalid. [Thus, any tear that occurs beneath the loose area called Anaf, invalidates the Tzitzis.]

Must one remove the Tallis?^[5] Yes. Background: If on a weekday, prior to wearing the Tallis Gadol before Davening, one discovered that his Tallis is invalid, it is forbidden for him to wear it even without a blessing. This applies even if he is in Shul [and is partaking in a Minyan].^[6] Nonetheless, if one only discovered the invalidation after already wearing the Tallis in Shul for Davening, then if he is embarrassed to remove it in front of those present, he may hurry and leave the Shul, and then remove it.^[7] If one is in private, he must always remove the Tallis as soon as the invalidation is discovered. In the event that one’s Tallis is invalid and he is embarrassed to Daven in Shul without a Tallis, he may Daven at home without a Minyan.^[8]]

May one put on a new Tallis in middle of Davening? Yes [see next regarding blessing], with some exceptions. Background:[9] If one received a new Tallis in middle of Davening then he may place the Tallis on in any area of the prayer, even in the middle of a Perek, with exception to if one is in the middle of the first paragraph of Shema, or is between Geula and Tefillah, in which case he may not even wear the Tallis.^[10]

Does one say a new blessing over the new Tallis? Yes. Background: A person must repeat the blessing upon wearing a subsequent pair of Tallis Gadol that he owns, and it is not included in his original blessing if he did not have it in mind originally or made a Hefsek in between.[11] This applies even if one borrows a Kosher Tallis Gadol from his friend after asking his permission is obligated to recite a blessing upon wearing it if it was evident to his friend that he borrowed it for the sake of performing the Mitzvah of Tzitzis.^[12] However, when using the Tallis of another person without his permission one is not required to recite a blessing prior to wearing it.^[13] Nevertheless, if one desires to say a blessing over it, he may do so.^[14] However, one who borrows the public Tallis of a Shul is obligated to say a blessing upon wearing it, just as is the law regarding one who wears a Tallis that he owns.^[15]

Where within the prayer may the new blessing be said? Between Yishtabach and Yotzer Or, or after Shemoneh Esrei. Background:^[16] One is not to recite a blessing over the Tallis during Pesukei Dezimra. However one may recite a blessing over the Tallis between Yishtabach and Yotzer.^[17] [Thus, in the event that the new Tallis arrived in middle of Pesukei Dezimra, one is to place it on without a blessing, and between Yishtabach and Yotzer is to then shake the Tzitzis and recite a blessing.^[18]] If one received the new Tallis after Yotzer but before completing the blessing of Goal Yisrael, he may place the Tallis on without a blessing and after Shemoneh Esrei he is to shake the Tzitzis and recite a blessing.^[19]

[1] Siddur Admur; Shlah Chulin 112a in interpretation of Rav in Menachos 39a; Elya Raba 8:10 in name of Taz and Shlah; Emek Haberacha 2:13; Soles Belula 12:2

Background: The Gemara states that if a string tore in the “Ikkur” or root of the Tzitzis, the Tzitzis is invalid. There are various interpretations of this word “Ikkur”. Does it mean the “Anaf”, the loose strings; does it mean the Gdil, the braided area; does it mean the area below the braid, which is the strings that pass through the hole? Admur in the Siddur, as well as the other Poskim listed above, take the approach that it refers to the braid and below the braid.  In the Shulchan Aruch however Admur takes the last approach, as will be explained. 

[2] Admur ibid; Rav in Menachos ibid

[3] Admur 8:13; Siddur Admur; Taz 8:8; Poskim ibid and in addition so also rule: Kitzur SHU”A 9:13; M”B 8:21; Biur Halacha 12:1 “Yeish Lismoch”; Tehila Ledavid 12:2; Chazon Ish 3:6; Kaf Hachaim 8:37; See Piskeiy Teshuvos 12:3

The reason: As when the tear occurs beneath the Gdil, by the holes, it is considered as if both sides have been cut off, and hence an entire string is missing. [Tehila Ledavid ibid]

[4] Siddur Admur; Shlah Chulin 112a in interpretation of Rav in Menachos 39a; Elya Raba 8:10 in name of Taz and Shlah; Emek Haberacha 2:13; Soles Belula 12:2

The reason: As whenever it tears beneath the Anaf, it is viewed that the entire string has torn and one hence only has three whole strings remaining. [Shalah ibid]

Ruling of Admur in Shulchan Aruch and other Poskim: In the Shulchan Aruch 12:6 Admur rules that only when a string requires a minimum length of 4 cm, is there a difference of opinion regarding if the string tore in the braided area. This implies that if the string does not require 4 cm, such as if the other side of the string still maintains a 4 cm length, then according to all [even Rashi] it does not invalidate the Tzitzis even if it tore within the braid. So rules also Biur Halacha 12:1 “Yeish Lismoch”; Kitzur SHU”A 9:13; Tehila Ledavid 12:2. This is all regarding if the tear occurred within the Gedil. If however the tear occurred below the Gedil, by the hole, then it is invalid according to all. [Admur 8:13; Poskim ibid]

Ruling of Mishneh Berurah: In Biur Halacha ibid, the M”B rules that if the tear occurred within the first set of loops, it is invalid. If it occurred after the first two knots made on top of the first set of loops, and there is 4 cm of string that protrudes, it is valid. [See Piskeiy Teshuvos ibid]

[5] Admur 13:5; Rama 13:3 [does not mention Davening]

Other opinions: Some Poskim rule that if there will not be a Minyan in Shul one may go to Shul and wear the Tallis without a blessing. [Reb Akiva Eiger on Michaber 13:3; Kaf Hachaim 13:10; See Tehila Ledavid 13:4 that it is implied from Admur that if there were to be a Gnaiy Gadol then one may wear it during Davening] See Piskeiy Teshuvos 13:4 footnote 36-37 for a full discussion on this opinion.

[6] The reason: Although it is embarrassing to Daven without a Tallis, nevertheless it is forbidden for him to wear it being that it is not such a great embarrassment to sit without a Tallis and we do not allow a small embarrassment to push off a Biblical command of the Torah. [Admur ibid; Taz 13:5; M”A 13:8]

[7] Derech Hachaim 8:2; Siddur Yaavetz;  Ketzos Hashulchan 7:11

Ruling of Admur: Admur ibid implies that during Davening the law takes a greater leniency by a Tallis Gadol, and one is hence not required to remove it right away, until he reaches a private area. This is inferred from a) Here he only mentions refraining from the initial wearing and does not explain what to do if the invalidation was discovered after wearing it. b) Here he writes there is “some embarrassment” associated with not wearing the Tallis during Davening while in 13:4 he states “there is no embarrassment” involved in removing the Tallis Gadol, and hence one must say there is a difference in level of embarrassment between the rest of the day and during a public Davening. [Ketzos Hashulchan ibid footnote 34; Tehila Ledavid 13:3]

Other opinions: Some Poskim rule one is required to remove the Tallis immediately in Shul and may not delay doing so until he exits the Shul. [Tehila Ledavid ibid in implication of M”A 13:8; See Piskeiy Teshuvos 13:4]

[8] See Reb Akiva Eiger on Michaber 13:3; Kaf Hachaim 13:10

[9] Admur 66:3 and 11; Michaber 53:3; Rama 66:2; Rabbeinu Yonah Brachos 15a; Rosh Brachos 2:10; Ben Ish Chaiy Vaeira 19; M”B 66:16; Kaf Hachaim 66:14

[10] The reason: As in the middle of the Parshiyos of Shema it is permitted to stop even to do Melacha, with exception to the first paragraph, as explained in 63:8. [Admur ibid; M”B ibid]

[11] Admur 8:21

[12] Admur 14:6; Rosh Chulin 8:26; Michaber 14:3; M”A 14:8; Ben Ish Chaiy Lech Lecha 5; M”B 14:11; Kaf Hachaim 14:14; Piskeiy Teshuvos 14:5 that so is the practical custom

The reason: Although a borrowed Tallis is not obligated in Tzitzis for the first thirty days [14:4] and hence its blessing is merely optional [14:5] nevertheless when one borrows a Tallis that is only worn for the sake of the Mitzvah, such as a Tallis Gadol and Tallis Katan, he must recite a blessing. The reason for this is because we presume that the lender gave it to the borrower on the premises that he is borrowing it in order to fulfill the Mitzvah and say a blessing over it. Now, since it is not possible to recite a compulsory blessing on a borrowed Tallis, therefore it is circumstantial evidence that the lender intended to lend it in a way that the borrower can say a compulsory blessing. Now, since the blessing is not compulsory unless it belongs to the person wearing the Tallis therefore it is legally assumed that the lender gave the borrower the Tallis as a legal present on condition to return after its use, and it hence belongs to the borrower until it is returned. [Admur ibid; Taz ibid; M”A ibid; Rosh Chulin 8:26; Tur Even Haezer 28] 

Other opinions: Some Poskim rule that a borrowed Tallis is never to be used, and if one has to use it, one is not to say a blessing over it. [Rashal Kol Habasar 53 [brought in Taz]; Taz 14:5; Chesed Lealafim 14:3; Mor Uketzia] In the Siddur of Admur it is evident that there are opinions that rule that one may not say a blessing on a borrowed Tallis unless it was explicitly given as a Matana Al Menas Lehachzir. [See Siddur Raskin p. 67]

[13] Admur 14:11 based on Michaber 14:4 as explained in M”A 14:8; So also rules: Toras Chaim Sofer 14:10; Kaf Hachaim 14:20; Az Nidbaru 6:44

Other opinions: Some Poskim rule one may not recite a blessing over a borrowed Tallis unless it was explicitly given to him as a Matana Al Menas Lehachzir by the owner. [Rashal Kol Habasar 53; Taz 14:5; See Reb Akiva Eiger ibid; M”B 14:11] Others however rule that when one borrows the Tallis it is always considered a Matana Al Menas Lehachzir, and hence a blessing must be recited. [See Shach Choshen Mishpat 358:1; Birchas Habayis 37:23; other Poskim in previous footnotes] Some conclude based on this, that one is to borrow the Tallis with explicit intent to not acquire it, and then not say a blessing according to all. [M”B 14:14 based on Derech Hachaim]

The reason: As the owner is not aware that one has borrowed the Tallis and hence it is not relevant to say that it is considered as if the owner has given it to him as a present on condition to return [in which case a blessing is required]. [Admur ibid; M”A ibid; see other opinions stated above]

[14] The reason: As although he is not obligated in the Mitzvah, being the Tallis is not his, nevertheless this is no less than the ability of women to say a blessing over a Tallis, despite them not being obligated in the Mitzvah. [Admur 14:5; M”A 14:8]

[15] Elya Raba 14:6 in name of Mordechai 950; P”M 14 A”A 6; M”B 14:7; Birchas Habayis 37:18; Chesed Lealafim 14:6; Derech Hachaim 9; Kitzur SHU”A 9:11; Mateh Efraim 581:14; Aruch Hashulchan 14:10; Toras Chaim Sofer 14:14; Ketzos Hashulchan 7:9; Ben Ish Chaiy Lech Lecha 5; Kaf Hachaim 14:15

The reason: As the Tallis of a Shul is considered owned by all Jewry, and hence is obligated in Tzitzis by any Jew who wears it, as he owns a part in the Tallis, and it thus similar to a Tallis of joint ownership. [M”B ibid] Thus, a blessing must be recited even if one has intent not to acquire it, as the Tallis is already his. [Har Tzevi 1:17]

[16] Admur 53:3; Michaber 53:3; Chayeh Adam 20:3; Gloss of Rav Akiva Eiger ibid; Kitzur SHU”A 14:8; Kaf Hachaim 53:7; Ketzos Hashulchan 18 footnote 23 based on the wording in Admur here “As it is forbidden to make an interval between Baruch Sheamar and Yishtabach.”

[17] Admur ibid; Michaber ibid

The reason: As it is permitted to make an interval in that area for the sake of a Mitzvah, as explained in 54:3 [ibid]

[18] Chayeh Adam 20, brought in Ketzos Hashulchan 18 footnote 23; Birchas Habayis 33:5; Hefsek Betefila 5:2-5 [p. 86]; See Admur 66:3

[19] Admur 66:3 and 11; Rama 66:2; Rabbeinu Yonah Brachos 15a; Rosh Brachos 2:10; Ben Ish Chaiy Vaeira 19; M”B 66:16; Kaf Hachaim 66:14

The reason one may not recite a blessing: Although by Tefillin a blessing may be recited, this is because Tefillin is an obligation to wear for Shema and Shemoneh Esrei in order to accept the complete yoke of Heaven. However Tzitzis is not an obligation of the body, as if one is not wearing a Tallis he is not obligated in Tzitzis, and he can read Shema without it being false testimony, being he is not wearing a Tallis that is obligated in Tzitzis. Therefore, it is forbidden for him to make an interval with its blessing even between the Perakim of Shema and its Brachos, and rather he is to wear it without a blessing. [Admur 66:3; M”B 66:16]Other opinions brought in Admur: Some Poskim [Michaber 66:2; Shivlei Haleket 15] rule one may wear the Tallis with a blessing between the Perakim. Other Poskim [Rabbeinu Yonah Brachos 15a; Rosh Brachos 2:10] however rule that only Tefillin may be worn with a blessing between the Perakim, however Tzitzis may not be worn with a blessing, for the reason explained below. Practically, one is to follow the latter opinion. [Admur 66:3; Rama ibid]

Other Poskim: Some Poskim rule one is to wear the Tallis with a blessing. [Michaber 66:2; Shivlei Haleket 15; Yechaveh Daas 6:4; See M”B 66:15 that a blessing may be said Bein Haperakim; See Shevet Halevi 3:15] Some Poskim rule one is to only wear the Tallis between the Perakim and only without a blessing. [Kaf Hachaim 66:15;]