I wouldn’t trust someone who doesn’t know who makes the iPod to replace the battery.
I sent in my iPod Mini for battery replacement a year and a half after I got it. They just send back another refurbished iPod of the same type, not your original one. This was a few years ago, so you might want to check if they changed this policy.
Certain iPod models (like the mini) had serious problems with battery life dying over time. What kind of iPod is this?
As a side note, my one trip to the apple store to address this very problem was quite unpleasant, and not very helpful. The staff there didn’t believe in responding to customers asking for help, unless there name was their computerized list. When I did finally get some help, they were impatient and not very helpful. Again, this was one store about 4 years ago, but I think you best bet is to send it back to apple or get a new iPod. I have had my new iPod nano for about 10 months and it has not once run out of battery power, and I use it almost every day.