12.6 with the engine running is a failed alternator. it should be form 13.5-14.5 volts at that point. with everything off, 12.6 is ok. since the alternator was replaced, you just might have it tested in a shop and also check the fuses to it. there should be one on the input side and then one on the output side. hopefully it is not a fusible link

Replace the alternator and have the battery load tested at auto zone.

Your alternator is not charging the battery! Normal alternator voltage with the engine running is 13.5 - 14.2 volts. The lower the battery voltage the dimmer the lights and slower all the electrical systems operate.

Who is the moron that put a thumbs down on this?????

A few months ago my battery light would come on occasionally...a few weeks ago I had my car serviced and they said my battery was dying (the light had not been on in months). I finally went to advanced auto and replaced (dodge intrepid...ie remove air cleaner housing, passenger tire, wheel cowl etc) the battery on Saturday and today the battery light came on and stayed on (I did not drive Sunday) when I used a dc volt meter the battery is reading 12.68 volts....with the car running the volt meter simply displays "1." is this a short to ground? The car is starting fine and driving fine...I drive an hour a day to work and its a new job! HELP! Could I have done something wrong?