Time for the Wein Cell to function
A Wein Cell MBR625 takes a bit of time after taking off the seal on the air holes to make the meter function. Unpack the battery, remove the seals, and leave the battery out for five minutes before installing in the camera.
And as mentioned, set the aperture ring on "A". Meter only functions there. $60 is a good price for a nice GIII showing clear lens and no cosmetic damage ($250 to 300 if it's a black model).
A seal kit is $10 to $15 and not a big job. The kit comes with ample instruction.
Also, before giving up on the Wein Cell, I'd question the mfr date of the battery, and time on the shelf on the Wein Cell you used. Better yet, test it with a voltmeter. They can be dead in the package when you buy them.
As a backup, get a NEW alkaline 625 from Radio Shack. The voltage is wrong (1.5 vs 1.35) and the discharge curve is horrible in a camera designed for the old mercury PX625, but you can test meter function.
I've had about ten of the GIII's and never seen one without any meter function at all given a fresh battery.