Replace Lithium Ion Battery Pack With AA Batteries

Alright I am going to start using a Pocket DVR Camera that I got a few years back once again. At the moment though the current batteries you get with the camera suck, and since it is a 2-3 year old camera by now I doubt there going to get better ones. Anyways onto the Technical Stuff..

The camera has 4 pins for the battery, the Positive and Negative pins are clearly labeled and I assumed the other 2 are for ground or something like that. The camera is looking for...

Rating: DC3.7V (Line with three mini dashes under it) 850mA

I already did the math and 3 AA batteries have an output of 3.6V while 4 AA batteries have an output of 4.8V. I think it would be easy and have the batteries last longer if there are 4 AA Batteries. Anyways what electronics can I put together to make a new battery pack for the camcorder. Sorry I do suck at electronics (good at soldering however) but I think it would be cheaper to build something that uses rechargeable AA batteries instead of spending $65 on a new battery that only last an hour.

On the side note, how would I calculate how long the batteries will last? I know the original batteries only lasted an hour since they provided 3.7V and had 850mAh.

Thank for the help, and also thanks for sticking though the stupidity.