Gemara

66a (Rabah): If the owner despairs, the thief acquires. We do not know if this is mid'Oraisa or mid'Rabanan.

(Rav Yosef): A thief does not acquire through despair, even mid'Rabanan.

67a (Ula): "You offer (an animal that is) stolen, lame or sick" equates a stolen animal to a lame one. It cannot become acceptable, even after despair!

(Rava): We learn from "his Korban" - not a stolen one. Obviously, he cannot bring a Korban before despair! Rather, it is after despair, for despair does not acquire.

68a: Rav Nachman and R. Yochanan hold that if a thief sold before or after despair, he pays four or five. We do not say that the thief acquired, and he slaughters or sells his own animal.

Rishonim

Rif (27a): The Halachah follows Rav Nachman and R. Yochanan, who obligate for selling before or after despair.

Rif (Kidushin 7a): If a man was Mekadesh with what he stole from others, she is Mekudeshes only if he already acquired it through despair. Then, it is like his.

Rambam (Hilchos Geneivah 1:12): If the thief has the stolen object and it did not change, it returns to the owner, before or after despair.

Rambam (Hilchos Ishus 5:7): If a thief was Mekadesh with Gezel and it is known that he acquired through despair, she is Mekudeshes.

Magid Mishneh: Even though despair alone does not acquire, here she acquires it through despair with Shinuy Reshus.

Rosh (7:2): The Halachah follows Rabah that despair acquires, but he was unsure if this is mid'Oraisa or mid'Rabanan. We are stringent to say that despair acquires regarding Kidushin, i.e. mid'Rabanan. Mid'Oraisa it does not acquire, like Ula, Rav Nachman, Rav Sheshes and R. Yochanan. Despair with Shinuy Reshus acquires mid'Oraisa.

Poskim

Shulchan Aruch (CM 353:2): If the owner despaired, the thief does not acquire. He must return the theft.

Beis Yosef (DH ul'Inan): The Rosh says that we are stringent to say that despair acquires. I did not find anyone who agrees with him.

SMA (3): Why didn't the Rema mention the Rosh's opinion? It seems that he relies on what he wrote in EH (28:1):

Shulchan Aruch (EH 28:1): If a thief was Mekadesh with Gezel and it is known that he acquired through despair, she is Mekudeshes.

Rema: If he was Mekadesh with Gezel after despair alone, she is Mekudeshes mid'Rabanan.

Chelkas Mechokek (4): This is from R. Yerucham, unlike the Magid Mishneh, who says that she acquires through despair and Shinuy Reshus. Perhaps the Rema discusses what he stole from her, like the Ramah cited in the Tur.

Rebuttal (Beis Shmuel 6): The Tur is unlike this. Perhaps even after despair, she can say 'I took what is mine!' Rather, he accepted to compensate her if it will be taken, or there was no Shinuy Reshus and it is Safek Kidushin mid'Rabanan. The Beis Yosef holds that a Kinyan mid'Rabanan makes Kidushin mid'Oraisa, unlike R. Yerucham.

Yam Shel Shlomo (7:7): The Rosh says that despair acquires l'Chumra (to be stringent). Perhaps it is a printing mistake; it is not in some texts. It should say 'despair acquires l'Chuda (by itself).' The Tur says that it is a Safek whether despair acquires mid'Rabanan, so we must be stringent about Kidushin. I.e. perhaps the Halachah follows Rabah. The Rosh was not unsure. He said that the Halachah follows Rabah, that despair acquires!

Suggestion: Perhaps the Tur had a different text of the Rosh.

Rejection (Yam Shel Shlomo): The Tur wrote in Kitzur Piskei ha'Rosh 'despair acquires mid'Rabanan. If he was Mekadesh, she needs a Get.' Rather, our text of the Tur is mistaken. Perhaps Safek refers to what he wrote afterwards about Stam Geneivah; it is a Safek whether there was despair. Regarding Gezeilah, the Tur wrote 'she needs a Get, like I explained about Geneivah.' He did not mention Safek, for regarding Gezeilah we assume that there is despair. In Siman 361, the Tur obligates returning the stolen object itself if there was no Shinuy, for despair alone does not acquire. When the object is intact, there is no enactment for penitents. Alternatively, he teaches the mid'Oraisa law. In Siman 362 he says that if the price of the stolen object increased after despair, the thief gets it, even though there is no Shinuy. The Rambam disagrees in this case. The Tur holds that whenever an enactment for penitents applies, e.g. the price increased, or so he will not need to toil (to return the theft itself), Shinuy (presumably, the text should say 'despair') acquires by itself.

Suggestion: Despair acquires only mid'Rabanan, so the Kidushin is only mid'Rabanan; if she accepted Kidushin from another man, we are concerned.

Rejection (Yam Shel Shlomo): Once Chachamim enacted, the money is totally his. Why does the Rosh say that the Kidushin is mid'Rabanan? All agree that it is mid'Oraisa, for when she gets the money there is also Shinuy Reshus! The Rosh holds that it is like Kidushin with others' property, since he did not acquire the money before he gave it. Still, she is Mekudeshes, for he acquires it when he gives it to her and she acquires. We learn from the opinion that obligates for selling a stolen animal only after despair, for then the buyer acquires. It is called a sale, even though the seller did not own it when he sold it, for the buyer acquires. It is no worse than a gift, for which one pays four or five. The Rif wrote Stam that despair does not acquire, i.e. even mid'Rabanan, yet he wrote that if he was Mekadesh, she is Mekudeshes, i.e. through Shinuy Reshus after despair. The Magid Mishneh (Hilchos Ishus 5:7) says so explicitly. However, the Rambam holds that despair acquires. The Rosh's opinion pertains to one who deposited the theft with Leah before despair, and after despair he was Mekadesh her with the theft in her hands. There is despair without Shinuy Reshus. According to the Rosh, through Chachamim's enactment she is Mekudeshes mid'Oraisa. L'Halachah, we must be concerned for the opinions who disagree with the Rosh. They rules like Rav Yosef because R. Yochanan, Rav Nachman, Rava and Ula hold like him. This is difficult, for they discuss the law mid'Oraisa! Perhaps despair acquires mid'Rabanan!

Rebuttal (Gra 4): All the Amora'im who say that despair does not acquire say so even mid'Rabanan. The Magid Mishneh's opinion is primary.

Bach (CM 353:3): The Rosh is stringent like Rabah, for it is not clear whether Ula and other Amora'im (who said that despair does not acquire mid'Oraisa) agree that it acquires mid'Rabanan.

Hagahos (in Tur ha'Shalem 353:6): The Bach holds that Kidushin after despair is only a Safek, for he holds (356:10) that when the buyer knew that it was stolen, Shinuy Reshus does not totally acquire.