The magnetoresistance of a conducting solid has been measured at high frequencies through the techniques of microwave spectroscopy. The measurement is based on the detection of electromagnetic power absorbed by conduction electrons under the conditions imposed by a resonant cavity and an external magnetic field. The magnetic field dependence of this power absorption is calculated and verified experimentally within one order-of-magnitude for a thin copper film. The method developed has potential application to the measurement of mobility of charge carriers in organic substances. In certain biomolecules, particularly those that are important in life processes, mobility measurements should be useful in distinguishing electronic and protonic conduction processes.