And what kind of faucet are we dealing with? Brand name might be helpful, but a picture might offer more insight still. Single handle or double handle?
I'll guess and say single handle. If that is correct, there are still separate seals inside for hot and cold water. After the cold water is shut off too, the faucet will have to be taken apart and the internal seals replaced. If it is a double handle faucet, the hot water side will have to be fixed, although I would do both sides while you are at it.
