I would bet input shaft bearing, like JimBlake said, its not the throw out bearing, because that wouldnt have a delay to the sound stopping.

And yes you can replace it without replacing the tranny, while you are doing it have the input shaft seal done as well, because usually when the bearing goes out, it takes the seal with it, and that will leak on your clutch and ruin it. I think i got quoted about 300 bucks to do it...but i dont remember if that was with the tranny already out or not, call around to tranny shops.