Dead iPod mini -- now what?

Hi, I have a 6GB iPod mini that suddenly went completely dead. Absolutely no signs of life on its LCD panel, even when plugged into a USB jack or a charger/speaker device. When I turned it on a few days ago I saw the "low battery" icon flash briefly, but it went dead immediately after that and has stayed dead. I am wondering if this means the battery has gone completely dead -- but if so, shouldn't it power up when plugged into an external power source? Or would a dead battery prevent even that from happening? How can I find out for sure what is wrong with it? I've seen battery replacement kits for very little money so I wouldn't mind giving that a try, but obviously if it is going to be much more expensive then I suppose I'd be better off getting a new iPod (though the mini meets all my needs so I'd rather not have to do that!)

Thanks,
Samizdat