Here's a link on how to replace the battery yourself. For all of about 30 bucks and a literally about 10 minutes of your time. Better than 60+ Apple wants to charge and two or three weeks (if not longer) for them to change the battery and ship back to you.
http://www.ilounge.com/index.php/articl ... ipod-mini/
Granted, that is for the mini, but they are essentially constructed in the same fashion.
The most difficult part I had was getting the battery, which more than likely you're going to have to buy on online.
It really is quite easy, you just have to be gentle.