Dodge Durango won't hold charge... it's not the alternator

I had an aftermarket stereo put in and that's when the issue started. The car
just won't hold a charge and I've tested the alternator to make sure that's not
the problem.

After jumping the car a few times I put a brand new battery in thinking that
maybe the battery was just old. But after about a week, the new battery had
totally drained. I took it into a Dodge dealership and they're not sure what the
problem is. We disconnected the stereo and that MAY have fixed the problem
but I have a question.

The battery I bought 4 months ago has since been completely discharged
about 5 times but it's still able to hold a charge when not connected to the car.
I just disconnected the stereo last week and hooked up the battery (which had
been sitting for a week) and the car started right up. Then I left the battery
connected to the car overnight and it was completely out of juice.

Could it have been that the battery was already so weak that the small draw
from the sitting car drained it or do you think that my problem is still there?
The battery is definitely weak, but it should still have enough power to stay
charged overnight hooked up to a car which shouldn't be drawing any power...
especially considering that it held a charge for a week while disconnected.
Thoughts?