Clearly quite a few dog lovers on this board. :thumbsup: No question that the best outcome would be for the dog to stay in the OP's loving home. But Deere Dude is unfortunately correct. The dog is legally still the property of the true owner, and until those rights are terminated, there is the risk that the owner will try to assert a claim in the future. At best, your family will lose the dog, and at worst it could get really ugly, as it did in the case of good friends of ours a few years ago. Their adult daughter took in a ragtag stray and nursed it back to health in their home. Weeks later the true owner learned about it and, wait for it, actually accused the family of theft and pressed charges. Family had to hire a lawyer to get the case dismissed.
So here's another approach for the OP to consider. Every local jurisdiction I know of has a procedure for dealing with lost and abandoned animals, and rehoming them when possible. Following that procedure will legally purge all prior ownership claims and convey clear title to the adopting family. So here's what I suggest Call the local municipal animal shelter and ask what their policy is for adopting out strays. Typically, it is to take a stray in, evaluate it and hold it for minimum period, say, seven days. If noone inquires about it within the holding period, it becomes available for legal adoption.
If this is indeed the policy in the OP's jurisdiction, I would take the dog into the shelter, say you found it in poor condition and nursed it while putting an ad on Craigslist that has gone unanswered. Leave your contact information and make it very clear that you want to sponsor the animal and adopt it if the true owner does not show up at the shelter. Call every day after that to ask how the little guy is doing. Heck, go in for a visit every day if you can make the time. Working in a municipal shelter can be very dispiriting for folks that love animals. I can pretty much assure you that the staff will be impressed by your concern and be eager to see that the dog ends up back with you after the mandatory waiting period. You may need to pay a modest adoption fee, but consider that it goes to a good purpose. And, you'll end up with documented clear legal ownership of the animal. You'll never need to worry about an unwelcome visit from a prior owner asserting a claim.