Won't hold the charge

I have an iPod Classic 5th Gen that will not hold the charge. When I plug it into a wall outlet, it turns on and functions but will not charge and dies immediately. When I plug it into the computer, it won't even turn on, as if there isn't enough power to charge it. I'm assuming it's the battery, but then do I lose my 2,500+ songs that is on it when I change the batter?! If I could just sync the iPod to my new laptop (the other was stolen) then I can just use a differ iPod. But, I don't want to lose all my music!