What would cause a new battery to keep going dead on a 2003 Dodge Neon?

The battery is only a storage device. The alternator replaces the electrical energy that the battery uses every time you start the engine and every time you turn on an electrical device in the car. If the alternator isn't working or isn't working properly, the battery will eventually discharge. Answer if the alternator is in fact working but you have let the battery run empty, simply getting a jump start might not allow the battery to charge sufficiently once you start driving again. you have to charge the battery up a sufficient amount or the battery will just run out again.