Saturn - SL2 :: Battery Power Comes Back On When Open Car Door?

So two or so months ago, I had the battery replaced in my 1999 Saturn SL2. About two weeks ago, something strange began happening, but only in the morning when I went to leave the house for work. I get in the car, turn the key in the ignition, and the car won't start--the clock and dash lights will come on for a split second, then go off. I'll try again, and this time the clock won't come on, the dome light won't come on--it's as if the battery is completely dead.

Here's the strange part: when I open and shut my car door, THEN the clock and dome light come back on. Same thing with my keyless entry: after I try to start the car and it won't start, keyless entry doesn't work. Then, I open and close the door and the keyless entry suddenly works again. So generally I have to go through this process of turning the key in the ignition and opening and closing the door for about 5 minutes. Then, the car will start as if nothing happened and it will be fine for the rest of the day.

No matter how many places I travel, how many times I stop somewhere and turn the car off, it always starts when I come back. It's only in the morning that it does this. Occasionally, I've been able to go 2 or 3 days without a problem (i.e., it WILL start in the morning). Today, though, it took a good 10 minutes to start, and I'm worried some day soon here it's just not going to start at all.