Close inspection showed that it was likely to be a broken wire inside the white cable, probably just by the microphone (the white cable looked thin and weak at this point - it had probably been bent to-and-fro too much).
This instructable shows some pictures of the steps needed to repair it.
Don't bother trying, unless you have:
- a fine soldering iron
- a sharp knife
- a strong magnifying glass
- a very fine pair of tweezers
- a steady hand.