1996 Honda Accord Car Cuts off

I started having this problem a couple of weeks ago. I was driving down the street, all of a sudden my car just cuts off and doesnt start. I am low on gas, not low enough for the gas light to be on, but still pushing about 1/8 a tank. I try it a coupld of times, the battery dies and I have to call my wife. I call my wife, she brings a gallon of gas and comes and gives me a jump.

The car starts fine and runs normally for a week. Then one cold morning, I start my car, it cuts off again. I start it again, drive about half a mile then it cuts off and the battery is drained. I take it to the auto store, where they check the battery and alternator.

The alternator is good, the battery is bad, so I buy a new battery adn drive for three days with no problem. Then as i'm driving down the road, the car cuts off. I start it up, it cranks, I drive about another 10 miles homeward bound, it cuts off while driving again but this time doesnt crank for about a minute. Finally I get it to crank and make it home. Any idea of what could be causing this?