Alternator?

[Elvis]

OK, here's the problem. Although I live in Texas I really don't use the a/c often because it is only a 5 minute drive to work so it hardly pays. Well, the last couple of times I used the a/c everything was fine for about 10 minutes and then the engine just died. Last night it happened again and resulted in a $70 tow truck charge because I killed the battery trying to get it going but it would just not fire. I've had the car - a '67 coupe with a 302 - for about 10 years and probably the only thing that I have not swapped out is the alternator. My thought is that the alternator may be failing and when it can't handle the extra load with the a/c on, it eventually kills the engine. Does that make sense? Thanks for any assistance. E.