The paper outlines the problem of the measurement of the Q-factors of resonators for the C.E.R.N. linear accelerator. The Q-factors involved are of the order of 105, and the resonant frequencies are in the range 200–300 Mc/s. The methods considered are briefly discussed, and the one which seems most suitable employs sinusoidal amplitude modulation. The accuracy depends on a low-frequency phase measurement, and, for this reason, accurate knowledge of the power law of the detector is not required. The modulation component undergoes a phase shift owing to the high Q-factor of the resonator. When this attains a magnitude of π/4 rad, the Q-factor is given by one-half of the ratio of the resonant frequency to the modulation frequency. Two alternative procedures are described. A mathematical treatment of the theory and some experimental results are included in Appendices.