I picked up a couple of non-functional GIIIs over time, and found much the same problem that Ken mentioned above - corrosion that went past the battery compartment. With one of them, the fix went just as Ken described - resoldered the wire to the (cleaned up) battery contacts. After replacing the seals (Interslice kit) and some cosmetic TLC, this camera became a dependable user and my favorite rangefinder - I use it regularly. The other one - not so much. The corrosion had followed the wire deep into the electronics of the camera. I corresponded with Jon Goodman, the Interslice guy, who has repaired a zillion of these cameras - he said what I have is a "donor" camera, at least as far as one that meters.
I know that there are a lot of folks that don't use the metering function of ther GIIIs, but I find that for the type of photography that I use '70s rangefinders for, mainly snapshots, a cycling camera and "street photography" (getting to think that description is way overused) the meter is very useful and, in the case of the GIII, quite accurate - even for E-6. Compose, focus and shoot.
The first thing I would do with a recently acquired GIII with a non-functional meter is check and clean the battery compartment. If that doesn't work, take the bottom off the camera and see the extent of any corrosion - if it's limited to the end of the wire, clean it up good and resolder. If the corrosion goes deep into the camera, I wouldn't spend a lot of time or effort into trying to fix it. Sunny 16 time!
I like the GIII so much, I bought the rather spendy (around $30 w/shipping) C.R.I.S. adapter for it, and get very long and stable service from the silver-oxide cell.