Friend Dave had ripped one of the wires on his Sony earbud headphones, MDR-EX38iP, so I decided to fix it. With the help of the Internet he now has a headset back in service.
The wires on these headphones are really very thin and fragile. There is no apparent physical shielding between the signal and ground wires, and twisting like wires together did not get the cut off earbud working. Luckily my friends at Instructables gave me hints to the solution.
Each thread of a Sony headphone wire is coated in thin plastic. You need to get through the clear coat to get an electrical connection. In this way the overall wire can be thinner because there is no need for a physical non-conductive barrier.
Insert heat shrink as desired. I used two layers for extra strength. Twist together like wires. Use a knife and carefully scrape the top layer of each wire. Slowly heat the wire with a soldering iron and solder a thin layer. The act of soldering will melt the clear coat and the solder will electrically connect the wires. Heat the heat shrink and you are done.
It is not as hard as it sounds, though you do need to be careful. The wires are very small and are easily damaged. Done.