Don't waste your time sending it back to Apple, it's a rip off.
I've got a 3rd generation 40gb iPod and when my battery started dying last year I found that I could buy a new one (for about £20-25) which had TWICE the capacity of the standard one already in the player. I used IpodUpgrade.net who were pretty efficient and I got the battery delivered in less than a week. Not sure what their shipping policy is like for the US though.
Anyhow, installing the battery was a piece of piss but I work with computers most of the time so I'm used to taking stuff like that apart. Seeing your beloved (and expensive) iPod laid out in front of you in pieces might be a bit daunting at first but so long as you're careful it should be fine. I've had mine apart three times and for extra help you can download a battery installation video which is pretty helpful.
The inside of an iPod is incredibly straightforward; circuit board, hard drive and battery... with both the hard drive + battery designed to be easily disconnected.
If that website can't help, look for a suitably high mAh capacity battery for your particular model on eBay but reference the right installation video when putting it in... honestly, it's a piece of piss! 
Mind you, if it's a hardware fault it's probably completely fucked :\ohwell: but if you get another iPod replacing the battery outside of Apple's service is definitely the way to go
Less money wasted!