iPod screen goes black, then turns itself off.

My iPod has been skipping songs for the past month or so, and, lately, it's started heating up while in use.

Earlier tonight, I started playing it after fully recharging the battery. A few minutes later, when songs began to skip, I reset it--and that's when the screen turned black. The iPod shuts off by itself after a few minutes, but pressing the "play" part of the click wheel brings the black screen back.

Is the battery dead, or is there something I can do to save it without replacing the battery? Attempting to recharge does nothing to fix the problem.

I've had my iPod since February 2005, if that helps.