DO you have conventional U joints on the front and rear of the shaft?

A 63 Belvedere should have the ball and trunion joint on the transmission end of the shaft. This is a good joint but a bit troublesome to work with, and find replacements, many have been swapped out to later U joints. Of course that requires a different transmission, with a slip yoke tailshaft. Some people have been able to eliminate the trunion joint by using Dodge truck driveshaft parts. I am going to guess thats whats in your car.

Ball and trunion pic: