My first suspect would be the voltage regulator. In the '60s and early '70s, these were external to the alternator, usually mounted on the back side of the radiator support. The most common failure point was a thin wire inside the regulator that would melt when the system was overloaded. You can check the basic workings by removing the cover from the regulator and looking for the remains of the missing connection. While you're in there, you can also check to see that the 'points' aren't damaged or welded together.
Other possibilities are corroded battery terminals, loose wiring connections, dirty or loose ground connections, etc.
I'm not sure if it will help, but here's a nice diagram and explanation of how the charging system works in GM cars from the '60s: