There are two ways a dog can get tape worms
If you are seeing what looks like "rice" in the dog's feces then what you are seeing are seqments of the adult worm being shedded.
Part of the tapeworm's life cycle is to live in the flea's gut. Thus a dog can get tapeworms by either eating the infected flea, or eating the infected flea's feces. My mom once had two dogs. One never got tape worms, and the other seemed regularly infected with them. The reason being the one that regularly became infected enjoyed grooming the other dog. So it ended up eating flea feces.
The best course you can follow is take worm puppy to the Vet for a tapeworm shot. Also they might give it Capstar to kill all indigenous fleas. Assuming your other dog if flea free it should be ok.