Nonreciprocal microwave devices, such as circulators, are useful in routing quantum signals in quantum networks and protecting quantum systems against noise coming from the detection chain. However, commercial, cryogenic circulators, now in use, are unsuitable for scalable superconducting quantum architectures due to their appreciable size, loss, and inherent magnetic field. We report on the measurement of a key nonreciprocal element, i.e., the gyrator, which can be used to realize a circulator. Unlike state-of-the-art gyrators, which use a magneto-optic effect to induce a phase shift of π between transmitted signals in opposite directions, our device uses the phase nonreciprocity of a Josephson-based three-wave-mixing device. By coupling two of these mixers and operating them in noiseless frequency-conversion mode, we show that the device acts as a nonreciprocal phase shifter whose phase shift is controlled by the phase difference of the microwave tones driving the mixers. Such a device could be used to realize a lossless, on-chip, superconducting circulator suitable for quantum-information-processing applications.