That is the classic noise of an alternator belt that isnt quite tight enough. My Neon is a 97, and it doesnt have an auto-tensioner. I have had to tighten the belt after the first few hundred miles, and again a month or two later. No more squeals.
If it has a tensioner that is automatic, it could be the tensioner is faulty, and either doesnt move smoothly, or has a bad bearing in the pulley. Im fond of replacing the tensioner pulley (and idler pulley if it has one) each time I replace serpentinve belts.