We introduce an all-fiber stationary phase shifter for a fiber-optic gyroscope (FOG) which simultaneously provides phase shifts of opposite signs in different cores of the dual-core optical fiber. We propose a new dual-core fiber-optic gyroscope (DCFOG) in which different cores of the dual-core optical fiber provide independent rotation rate measurements. The device enables implementation of a differential scheme, which ensures the stability of the measured phase shift. As a computer simulation result, the accuracy of the rotation rate sensing is increased by up to 10 times at typical noise levels.