R1 Alternator question

[Paul Parker]

The '63 R1 I inherited from my dad, an electrical engineer for NASA who bought it new, no longer charges the battery.   What should be my first step?  

A tiny bit of background;  when I got the car 8 years ago, it ran great.  AC blows cold, everything works as  it should.   Then it sat too long.    The horn and interior lights quit working.   Otherwise, it still runs beautifully until the battery dies.    Now the lack of charging.      Could something other than the alternator be the issue?   

Dad moved the battery to the trunk and put a fake up front. 

Do people still rebuild alternators, as I saw my dad do for countless other cars and maybe even this one?   I'm curious about that Wilson sticker...

Paul in Sarasota, FL