I'm on my 4th MX1000, 3 were replacements for failed units from connection problems with the base station/charger that would cause skipping and moments of inoperability. I have noticed that Logitech gear lasts about a year. I called Logitech and told them the problems that I was having, which seems less than what you are having, and they sent me replacements with relatively little trouble. Same thing with the Apple cordless keyboard. After X amount of time both would loose the ability to connect and/or operate properly, thus, needing replacement and both at about a year into their use.
I trust you have tried to reconnect the mouse to the Charger/Receiver and/or reinstalled the LCC software for your OS?
I have also noticed that certain versions of the LCC software will cause problems with OSX and/or will not operate properly. But in these cases the mouse would still operate under the OS mouse pref parameters to the point that the health of the mouse would let it.
I was told once that there was a problem with the charger stations for the MX1000. That is the first place for connectivity.
Call Logitech and talk to them. This their problem and I doubt that you will find any solutions here. But a good trouble shooting activity, if you can, is to try it on a PC. If it works there then the mouse and parts are in working order. I have noticed that Logitech is PC first and foremost, even though they say their gear is Mac compatible.
G4 FW800, 2GB RAM, 2-DVR110s, 4-500GB HDs, 3-Monitors, 2-Turtle Doves, 1-iTTUSB Mac OS X (10.4.8) Totally Stuffed/Max'd Out with lots of great toys.