Use arc welding. For those that like to clean the brake and shifter cable, the fray ends give some problem. I have tried soldering, super glue but the best is arc welding. I have also used acetylene welding but that is cumbersome. Here is how I do it.

Clamp the ground to an area of the brake that can take a little char, like the hex nut on the brake pad. Bring the electrode to the wire which will cut it. The cut is not usually clean. To get a blunt end, bring the electrode directly to the end. You will get a very clean and blunt end! Put a piece of card board to shield spark from damaging your paint\tire.

I live in San Jose, CA. I will do it for free if you are in the area.

For those that tried soldering. Here is another tip. In order for the solder to stick, you need to use an acid with a little of tin mixed in to clean before soldering. The best way to get some tin is from a used battery. Break up a C or D cell battery; discard everything, except the outer tin shell. Put about 2 square inches into 1 cup of acid, any strong acid will do, like H2SO4. You don't need that much acid for this job. I paid 7 dollars for the tip of where to find a source of tin from a guy in Vietnam.