Electrical charging system question, Alternator? Options

[rjames]

I installed a brand new battery and a new voltage regulater last week in the car. My volt meter is showing 12.5 volts at the terminals. With the car running and the lights and fan on my volt meter fluctuates between 12.7 and 12.9, then slowly starts to dip, getting as low as 12.5 before I decided to shut the engine off because I could see that the battery was slowly draining.

Would it be logical that the alternator might need replacing? The alt. belt also seemed tight (not sure exactly how tight it should feel though.) I can check the electrical connections (have to find them first though.) The connections at the battery all look good though.

It doesn't look like a fun job to replace the alternator, but I'm up for the challenge. Anything else I should look for before I go this route??

Thanks!

PS: Forgot to add that the alternator light on the dash comes on when starting the car, but goes off right away once it's started and doesn't come back on at all when driving.