The most promising type of micro-optical gyro is a resonator gyroscope. At the same time, the principle of operation of all prototypes of resonator micro-optical gyroscopes developed to date involves scanning a passive ring resonator in frequency. We devoted this work to a new approach to the construction and operation principle of a resonator micro-optical gyroscope. This approach does not require scanning a passive ring resonator in frequency and is realized through the use of a Mach–Zehnder modulator with a passive ring resonator connected to one of its arms. It was shown that, with this approach, the angular velocity can be determined, first, by the difference between the maxima and minima of the radiation power at the output of the Mach–Zehnder modulator when the optical length of one of its arms is modulated and, second, by the displacement of these maxima and minima. A common drawback of all considered angular velocity measurement methods utilizing the Mach–Zehnder modulator is the limited measurement range. For the resonator considered in this paper, the measurement range is of the order of ±1500  rad  /  s.