Acquire the proper connector for the microphone. If you plan to plug it into a modern radio, it require either an eight- or four-pin plug.
Remove the screws from the base of the microphone, and trace the individual wires within the cable. There are three wires, but the colors may vary, depending on the model of microphone and the year it was produced. The ground wire is wired to the shield of the cable. Another will be wired to the "Talk" switch of the microphone, and the final wire is connected to the microphone diaphragm.
Re-fit the microphone base and replace the screws. Use the cutters to snip off the current proprietary or outdated connector. Strip off about 5 mm of the insulation from the end, and separate the inside wires.
Heat up the soldering iron, and unscrew the rear conical portion of the connector to expose the contacts. Thread the conical section onto the wire, allowing you to slide it back up to the rest of the connector once the wires are soldered to the contacts.
The talk-switch, ground and microphone wires should be soldered to pins six, seven and eight, respectively on an eight-pin connector. On a four-pin connector, the wires should be connected to pins two, one and three respectively.