I had the disconnection problem with both rechargeble and alkaline batteries.
I managed to fix the latter with the following (from a poster on this site) -
1) Open Bluetooth Preferences
2) In the left hand window, select 'Apple Wireless Mouse'
3) Beneath the left hand window is a small gear wheel icon. With Apple Wireless mouse still selected, click on this and select 'Add to favourites".
This cured the problem for me as far as alkaline (non-rechargable) batteries are concerned but I still need a small amount of BlueTak on the battery compartment over each battery to stop them coming loose.
Rechargeable batteries are something else!! I gave up trying to get these to work. With these my Magic Mouse simply didn't light up at all. I assumed that as they were 1.2v not 1.5v that this was the problem.
Then I noticed that Apple Store was adiing people that they would work. I queried this elsewhere on this site and it's possible that they will work but for a small difference in the design of the cans at the + end.
http://dl.dropbox.com/u/11373233/MacDiscussions/Magic%20Mouse%20folder/Battery%2 0Tops1.pdf
I'm about to try the silver foil at the - end dodge that was mentioned earlier in this thread. If that works then I've got it sussed.