Short answer. You do not have to hand over your side of the exchange. Article 1639 of the Civil Code says a party who received a thing in barter and proves it did not belong to the person who gave it cannot be compelled to deliver what they offered in exchange, and is entitled to damages as well.
What the law says
If one of the contracting parties, having received the thing promised him in barter, should prove that it did not belong to the person who gave it, he cannot be compelled to deliver that which he offered in exchange, but he shall be entitled to damages.
Civil Code, Article 1639 — Barter: Thing Not Owned by the Giver. Read the full provision →
A defence you can raise before delivering
Barter, or exchange, is a contract where each party gives a thing rather than money. Article 1639 protects the party who has already received but not yet delivered. If you can show that what you were handed did not belong to the person who handed it over, the law will not force you to complete your side. You keep your own property and the other party cannot sue you for specific performance. Crucially, the burden is on you: the article requires that you prove the defect in ownership, not merely suspect it or hear about it.
Damages come on top of the defence
The article does not stop at excusing you from delivering. It expressly entitles you to damages, because a failed exchange is rarely costless. There may have been transport, registration or notarial expenses, opportunities passed over, or a business that was arranged around the thing you were promised.
You will need to prove those losses in the ordinary way, with receipts and records. The entitlement is against the party who gave you something that was not theirs to give; it does not run against the true owner, who has done nothing wrong and is simply recovering what belongs to them.
What happens to the thing you received
The defence lets you refuse to deliver, but it does not make you the owner of what you were given. A person cannot transfer more right than they have, so the true owner can generally recover the thing from you, and you should not sell it, consume it or improve it heavily while the question is open.
In practice the sensible sequence is to stop performance immediately, notify the other party in writing of what you have discovered, hold the thing without using it, and take advice on whether to return it to the true owner or to deposit it. Barter of registered property should never be completed without checking the certificate of title and the registry records first — that check is far cheaper than this remedy.