First gen iPod touch battery drops suddenly to 10-15% charge.

My first gen iPod touch, after about six hours disconnected from a power source (even in sleep mode) drops to 10-15% charge suddenly. It'll be 95% charge and just die like that. It's getting annoying to the point I've almost entirely stopped using it, because it's just a hassle to carry it around and have to fully turn it off and back on whenever I want to use it, and still, only six hours of battery life doing nothing.

I have an iHome, and it seems that people have trouble with them and battery life, but no cases I found matched mine.

Is it possible to get it replaced at a discount? I paid $400 for it, I really don't want to have to spend another $300 for a new one.