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I've had mice replaced under AppleCare for my PowerMac G4, though that was a few years ago. I would assume it's worth a try, at least.
 
I'm assuming you're talking about the little scroll ball on the Apple Mighty Mouse. I've heard that when this happens it needs cleaning. If that doesn't work, yeah you can get a replacement or it repaired through AppleCare.

Everything that came with your computer (in the box) from Apple is included under the AppleCare.
 
This happens with my mighty mouse as well. Get a slightly wet paper towel and wipe the trackball about 10 times with the towel. (Just use your pointer finger and move the trackball back and forth like you would normally do). This usually works for me.
 
Thanks.

How do you clean it?

cut a thin piece of paper, feed it in and turn it around in the scrol ball hole, slightly dampen it, the scrol ball getting stuck is a common problem, easily cured in 2 shakes of a lambs tail ;)
 
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