If you found it outside, you need to return it to its nest. A baby rabbit without the care of its mother is not likely to live very long. Either way, you really need to call a vet. The rabbit needs the care of a professional.
I have to agree with your mom though, wild rabbits are not meant to be pets. Whatever happens, you should put at least the healthy one back where you found it, unless you are absolutely sure their mother is gone.
Katie is correct, if this is a wild rabbit, it needs to go to a wildlife rehabber. Second choice is returning it to the nest. Not sure its legal for the vet to treat a wild rabbit, although they may kindly put it out of its misery. Chance of survival is pretty much nil. With a rehabber, it may have a ghost of a chance. And sometimes mom's can work miracles. Unfortunately you can't. To hang on to the rabbit is cruel, you're just another predator stressing it out.
ETA: If this is a domestic rabbit, a vet can treat it or put it out of its misery. If you can't go to a vet, but them back with mom and hope for the best. But the bacteria in a cat's mouth is lethal to rabbits. Not a great outlook here.