It sounds like an odd problem. How do you know it stops charging when driving? Will the new battery go dead after a few hours of driving? A little history of how and when this problem developed would be useful. What steps have you taken so far? Any recent work done?  Install a temporary meter so you can monitor the system as you drive and see what the amps are during different conditions. Dont trust the amp guage in the vehicle and especially if its just a light. Do you have an external regulator? If so, that could be the problem. Also check all electrical connections at the starter and solenoid. If these are loose or any broken terminals, it may cause an open circuit when the engine is under load and it shifts enough to put tension on the wires. As long as your at it, check for broken motor mounts that will let the engine move alot and pull on wires. Im just guessing but you havent given much to work with here. I will watch this question to follow up. Good luck.