I've just bought a brand new, sealed iPod Nano 3rd Generation iPod. Will Apple install a new battery instore for me?

I've just bought a brand new, sealed iPod Nano 3rd Generation from eBay. However, its not charging up properly. I've tried charging by my dock, wall-socket and computer. Still shows battery to be low and says it needs charging. It was set up OK on iTunes and I put some albums on it. It also plays fine whilst on my dock. When not attached to anything it just keeps powering off. Can only think as this Nano was manufactured in 2007 the battery has ran out of juice and needs replacing. My question is, is will Apple replace this instore for me? If so, can I just go into one of their shops or do I have to make an appointment? I have 2 stores in Bristol I can visit. Many thanks.