Unable to connect Galaxy S3 to PC via USB
Update: Try cleanup and reinstall of drivers as mentioned here. Seems to have worked for others.

This seems to be a known issue, and many have had success by using a different cable, connecting to a USB 3 port etc, but clearly, those won't work in your case.

Not sure what your usual usage is, but for file transfers, you can use Airdroid, which connects through wifi.