Bathing Mighty Mouse

I drive a Mac G-5 iMac. Recently, I added a multi-function, Mighty Mouse™. I love it. Or, I did until the scroll ball stopped working.
But not to worry, I wiped it with a damp cloth and it works again, sorta.
My question is, how do I clean the darn thing properly? I know the ball is pulling dust and crap inside, and transfering it to the contact inside, and wiping it off will only work for a while.
I know all this, because I’ve been cleaning Mac mouses since 1995. The only one that didn’t do what I just described was the single function optical I gave up for this one.
The ball is the same material as the old ones, just smaller.
Help me, please.

I’m not familiar with the brand you’re talking about, but is it different than an ordinary mouse in that it lacks a little holding ring on the bottom that twists off to let the ball fall out and give you access to the rollers?

It is quite different than PC mouses. May I present Mighty Mouse™. The ball I’m talking about is the one on the top.