1989 Olds 98

I recently bought an Oldsmobile 98 Regency. I love my car, but the only thing that I have noticed that drives me absolutely crazy, is when I am going down the freeway at around 60 to 65 mph, the car starts to shake and vibrate, even causing the passenger seat to vibrate. I can slow down to below 60 and it seems to stop. I have had my tires balanced, and this did not fix the problem. If someone could please tell me what might be causing this problem, it would be greatly appreciated. Please e mail me.
Thanks,
Nick Clark
Phoenix, Az