Starts up fine, but after I drive 10-20miles the car lights start getting dim?

My 99' Honda Accord starts up fine, but after I drive 10-20miles the car lights start getting dim (dash and headlights) and then the car dies. Initially thought it was bad alternator but after I replaced that, car did exact same thing. When I first purchased the vehicle (2 months ago) it was stalling out while I was driving, but would start up again, and I wouldn't experiance the problem again for days at a time. If it is some regulator problem would/ could this be why I experianced my stalling problem? I don't know much about cars, so I was curiousif the ignition switch has any effect on the alternator performance, or lack there of. Hope you guys might have a couple ideas on what this cars issue is! Thanks