This was a troubled phone. The charging port worked, but there was no access to data on the unit; and the data partition was corrupted, so that I couldn't get past the file system check no matter what boot method (standard, recovery, safe) I used. So this system couldn't even be rooted.
The Galaxy SIII is beautifully designed for repairs -- which is to say, it's put together with screws instead of adhesives. This fix was not much more difficult than replacing the battery. Really.
Unlike, say, early iPhones, all ten screws seem to be the same size, so there appears to be no need to keep track of which screw came from where. I kept track just in case. (There is an eleventh screw that is obviously different.) After the repair I was surprised to see that previously downloaded ringtones were still present, because I assumed that all the system's memory was on the motherboard.