As a long-term reader of alt.folklore.urban and the Jan Harold Brunvand books, I knew this was a variant of one of the oldest urban legends of all - in fact, it's "The Mexican Pet", the title of Brunvand's 1986 book. He collected a similar story "in the wild" in 1983. Snopes has the details from the book and more:
A woman from La Mesa, California, went to Tijuana, Mexico, to do some shopping. As any visitor to this border town knows, the streets near the shopping areas are populated with stray dogs. The woman took pity on one little stray and offered it a few bites of her lunch, after which it followed her around for the rest of the afternoon.
When it came time to return home, the woman had become so attached to her little friend that she couldn't bear to leave him behind. Knowing that it was illegal to bring a dog across the international border, she hid him among some packages on the seat of her car and managed to pass through the border checkpoint without incident. After arriving home, she gave the dog a bath, brushed his fur, then retired for the night with her newfound pet curled up at the foot of her bed.
When she awoke the next morning, the woman noticed that there was an oozing mucus around the dog's eyes and a slight foaming at the mouth. Afraid that the dog might be sick, she rushed him to a nearby veterinarian and returned home to await word on her pet's condition.
The call soon came. "I have just one question," said the vet. "Where did you get this dog?"
The woman didn't want to get into trouble, so she told the vet that she had found the dog running loose in the street near her home in
La Mesa.But the vet didn't buy it. "You did not find this dog in La Mesa.Where did you get the dog?"
The woman nervously admitted having brought the dog across the border from Tijuana. "But tell me, doctor," she said. "What is wrong with my dog?"
His reply was brief and to the point. "First of all, it's not a dog — it's a Mexican sewer rat. And second, it's dying."
I guess in the modern variant, the man actually pays for two of them. No one seems to be saying how he found out his weasels (or ferrets, or giant rats - the reports vary) were not actually dogs.