2002 Ford Excursion

I have a 2002 ford excursion 5.4 gas I just replaced the battery as it went dead on me tonight 7/25. Since then my battery/alternator light is on and wont go off. I put a voltage meter on the battery terminals and have about 13 when I turn the air, lights, and rear defog on it drops to 12.2 or so. Does this mean my alternator is bad, if not how do I get the bat. Light off?
July 25, 2007.

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