Gemara

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

Perhaps it is like an Aveidah. If the owner despaired before it was found, the finder acquires it mid'Oraisa. The same applies to a thief;

Or, perhaps an Aveidah is different, for the finder was allowed to pick it up. A thief acquires only mid'Rabanan. This is an enactment to help people repent.

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

66b - Question (Abaye - Mishnah): Hides of a Ganav (covert thief) can receive Tum'ah through intent (to use it as is, for a table-cover). Hides of a Gazlan (open robber) cannot.

R. Shimon says, hides of a Ganav cannot receive Tum'ah through intent. Hides of a Gazlan can, because the owner despaired.

Rebuttal (Rava): A change in the name of the object is like a physical change.

One acquires through physical change, e.g. he stole wood, and now it is Kelim. Similarly, it was called a hide, and now it is called a table-cover.

67a (Ula): "You offer (an animal that is) stolen, lame or sick" equates a stolen animal to a lame one. Just like a lame animal cannot become acceptable, also a stolen animal, even after the owner despairs!

(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): If a thief sold before despair, he is liable (four and five).

He learns from "and will sell it" - whether before or after despair.

(Rav Sheshes): He is exempt. (He pays only Kefel).

Rav Sheshes holds that he is liable only when the sale is valid, i.e. after despair, like liability for slaughter, in which his action accomplished something.

(R. Elazar): A Ganav who sells pays four or five. This implies that we assume that people despair when something is covertly stolen.

(R. Yochanan): We can learn from kidnapping. One is liable without despair (the victim does not despair from escaping)!

Inference: R. Yochanan holds that one is liable (four and five) before despair.

R. Yochanan: One is liable also after despair.

(Reish Lakish): He is exempt. The thief acquired; he slaughters or sells his own animal.

114a (Mishnah): If tax collectors or thieves took Reuven's donkey or garment and gave him a different one, Reuven may keep it, because the owner despairs.

Rishonim

Rif (27a): Rav Sheshes holds that one is liable for selling only when the sale is valid, i.e. after despair. Rav Nachman and R. Yochanan obligate for selling before or after despair. The Halachah follows them, for the Halachah follows Rav Nachman in monetary laws, and R. Yochanan holds like him.

Rambam (Hilchos Gezeilah 2:1): If the stolen object did not change, rather, it is like it was, even if the owner despaired, and even if the thief died and his children have it, it returns to the owner.

Hagahos Maimoniyos (1): Rav Hai Gaon permits to buy from a Gazlan, because despair acquires.

Yam Shel Shlomo (7:7): The Gemara (119b) forbids buying from a Ganav or Gazlan! We must say that this is when there is no despair, i.e. before the owner found out that it was stolen. One with Yir'as Shamayim should refrain from repulsive things, for the thief must fulfill his Mitzvah to return it. There is no enactment to help penitents as long as he has the object intact. One who buys from a thief supports his Aveirah and encourages the thief to steal again.

Rosh (7:2): Rabah and Rav Yosef argue about whether or not despair acquires. Rabah holds that it acquires, but he is unsure if this is mid'Oraisa or mid'Rabanan. Rav Yosef holds that it does not acquire, even mid'Rabanan. We hold like Rabah. We are stringent to say that despair acquires regarding Tum'ah of stolen hides and Kidushin, i.e. mid'Rabanan. Mid'Oraisa it does not acquire, like Ula, Rav Nachman, Rav Sheshes and R. Yochanan.

Poskim

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

Yam Shel Shlomo (7:7): The Rosh brought a proof from hides, for this was Rabah's primary proof.

Mordechai (68,69): We conclude that despair does not acquire. This refers to theft. If one finds an Aveidah after the owner despaired and said 'woe, for my monetary loss', since it came to his hand b'Heter, he acquires through despair. However, Rav Hai Gaon says that a thief acquires through despair, and one may buy from him what he stole. He relies on the Mishnah (114a) that says that if tax collectors or thieves took Reuven's donkey and gave him a different one, he may keep it, for the owner despairs.

Question (Pnei Yehoshua 66b Tosfos DH d'Gazal): Regarding Tum'ah of hides, the Tana'im argue about whether there is despair with a Stam Ganav or a Stam Gazlan, but all agree that despair acquires mid'Oraisa!

Answer (Tosfos 67a DH Ha): There, he acquires due to Shinuy ha'Shem, not due to despair. The despair merely enables his intent to make Shinuy ha'Shem.

Pnei Yehoshua: Reasoning says that Tum'as Mishkav should not depend on who owns it. We limit the Chidush to exclude a case where the owner intends to get it back from the thief in Beis Din, but not if he despaired.