Although there are many quick fixes for broken car wires, soldering is simple and by far the most lasting solution. Soldering a wire requires a few supplies and tools: a soldering iron, wire strippers, PVC heat-shrink tubes and rosin-core solder designed for electrical wiring, typically sold as "60/40." Creating a good solder joint takes practice, so consider practicing on scrap wires before making the repair.
Begin by stripping a half inch of insulation from each side of the broken wire. Slip a heat-shrink tube onto one side of the wire before soldering. Carefully twist the two ends together. Make sure the soldering iron is hot, and then heat the wire joint from the bottom. When the joint is thoroughly heated, touch the rosin core solder to it. Only a small amount of solder is needed to bond the wires. Avoid applying solder directly to the soldering iron's tip.
After the joint is complete, the section of bare wire needs to be shielded. Slip the heat-shrink tube over the bare section. It is best to use a heat gun or hair dryer set on high to shrink the tube, but a carefully directed lighter also works. Electrical tape is an alternative to heat-shrink tubes, but it does not provide as much rigidity and is a poor choice for thick-gauge wires.