'64 Chevy Dead

My '64 Chevy was running excellent. I pulled into a gas station. It would not restart. There were no head lights, nothing. Completely dead (everything). I had it towed home and and tried restarting, and, of course, after paying the tow truck guy, it started right up. The next evening, I took it out for a short run. I pulled into a parking lot driveway when suddenly, engine and lights quit at the same time. I was able to coast into a parking spot.

Items I have checked:
Battery- reads 12v, terminals looked good but I cleaned them anyway.
Battey cables- checked for continuity and OK.
Tested voltage of battery cable at starter. This was OK.
Tested ground cable to block- this seemed to be OK.

I do have an alarm system, but when activated the ignition switch is killed, but all lights still come on.

Any clues would be helpfull. Thanks!