Good Kitty!

This morning I was awakened by the scuffling sounds of a cat
running around and playing. That doesn't happen too often
anymore with Cliff. He is 12 years old and just doesn't run
around like he once did. I started telling him to stop, then
realized he was actually chasing something.
A mouse.
He literally had it pinned behind a couple of presentation boards I had leaning against the wall in the corner of the bedroom. So I pushed a heavy pair of shoes up against one side to make sure it stayed there while I went to look for some way to get it out of there. I guess I pinned it too hard because by the time I got back in there with my silicone oven mitts and a container, it was dead (I think it must have suffocated). Well, that's one way to deal with it. As disgusted as I am by it, I do feel kind of bad for it. Oh well.
But now something my mother has always said about mice comes to mind, 'If there's one, there's 9 more waiting back at the nest!' So I am left wondering how to deal with them. I want something that is
a) humane (or at least kills them quickly)
b) environmentally sound and
c) not a trap from which I will have to clean the bodies (traditional mouse traps give me the heeby jeebies). Any suggestions? How do you deal with mice?

Oh, and by the way, J, I told you he was a good cat!!!