UPDATE:

Scotty/blowngp: So I started it up my car this morning and it starts up fine. So it definitely looks like the alternator/voltage regulator is kaput.

I don't know if there's a way to replace only the voltage regulator, but I guess I'll pull it off and have a look. Thanks for all your help with this guys!

Scotty, Good luck with the lawnmower. Hope you find it to be a cheap and easy fix!