Answer

In principle, since it came into his possession unlawfully (that is, before despair), he has an obligation to return it to its owner like any other lost item.
On Wikipedia, under the entry “Unknowing Despair,” the opinions of the halakhic decisors regarding use of the item in the meantime are summarized:
The medieval authorities (Rishonim) disagreed whether one may use an item he found,[29] or whether “it must remain until Elijah comes,” and he may not use it at all.[30]Those who maintain that as long as he does not know to whom the lost item belongs, he acquires it and may do with it as he wishes, explain that the ruling that unknowing despair is not despair is relevant only in a case where the identity of the owner is known.[31] As a practical halakhic ruling, Rabbi Ben-Zion Abba Shaul[32] ruled that one may use it.
In a case where someone took an item from its owner and he knows clearly that the owner agrees that he should take it, even though he did not receive explicit permission, in such a case, according to the Beit Yosef, even Abaye agrees with Rava.