92 LeBaron goes dead and silent

I'm lost on this one. Car runs and drives seemingly great. We will turn the car off and later try to restart and everything is dead, no lights interior or exterior. No clicks from starter no dash lights, etc... wait 20 min. or next day, everything lights up and car usually starts. However, the battery seems to drain rapidly. I took the alternator to a 40 year experienced rebuilder and he says it's good.
Where do I go from here?
Thanks one and all, Mac.