Stick a tampon in the finger and thumb holes, then attach the end to some Silly Putty that completely covers them (one glob per hole). Put the ball in the dishwasher. MAKE SURE YOU TURN OFF ANY HEATED DRY CYCLES!!!!! Turn off any "sanitizing-rinse" wash cycles or anything too hot, like heated dry cycles.
I run the machine without any other dishes in there because the oil and garbage coming off the ball flies into them. But then you only need a light cycle.
Take it out, wipe it down, and you should be good to go. It will feel as tacky as a basketball when you're done.