New Battery Doesn't Stay Charged - Alt Checks Out Fine on 1993 Nissan 240SX

My battery suddenly died a week ago. AAA came out, did a diagnostic, and said that the battery needed replacing. They also checked the alternator and electrical system and declared them fine. Now it has happened again. The new battery is completely drained less than 7 days in. My only clue is that battery light on the dashboard does not illuminate at all. The rest of the indicator lights illuminate. Could this be telling me something? Does it sound like a big deal or a minor fix? Any information appreciated.

Asked by for the 1993 Nissan 240SX
If the Battery light does not illuminate, your battery will never charge. This circuit must work for you battery to charge. It could be just the bulb or a fuse or wiring. When you check a charging system, you must verify that the charging system bulb circuit works by turning the key on-engine off and the bulb should light up. Then you turn on the engine and the bulb should go out in 1-2 seconds ( or less ). If this doesn't happen, your charging system will not work properly.
How to fix this problem?
If the bulb won't light, it is often just a worn out bulb. Do that first, namely , replace the charging indicator bulb.

1 more answer
There must be something slowly draining the battery if everything checked out OK, Check things like the glove box door light, trunk light, radio etc or it could be a short somewhere in the vehicle, probably needs a technician thats good at trouble shooting electrical problems to take a look at you vehicle if you can't find anything left on.