Forget what these guys said. Tape 'em up.
First, do 6 or so inch piece of tape around the middle of the thumb, which should get you about two times around, and don't pull it too tightly if you want to easily bend your thumb. Then, with a shorter piece, tape over the end of the thumb, slightly overlapping the first piece of tape so it still extends a little past the end of the thumbnail, and pinch it closed at the end.
I've also been told that taping the end of the middle finger makes the grip even more secure, but I'll admit I've never tried it (although my wife has, and found it to be true).
Don't mind the band-aid. That's from when I tried to cut the end of my finger off when peeling fruit to make a cocktail.