If you want to do it yourself, there are online guides including this one
A 20GB iPod is probably 4th gen, but it could be 3rd or even 2nd gen. There are probably YouTube videos showing the procedure.
You can get the replacement part on eBay and elsewhere. Just make sure you know your iPod model, so that you get the right battery.
Most replacement batteries come with a pair of the plastic/nylon tools seen in the guide.
There are also repair businesses that will do the work, if you don't feel comfortable with what you see in the guide.