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The optical Kerr effect in a reciprocal fiber-optic Brillouin ring laser gyroscope is investigated. It is found that the Kerr effect, as the major error source of the rotation rate measurement in this gyroscope, causes a bias of beat frequency with nonlinear dependence on rotation rate. A differential Stokes power of 1 mW inside the ring resonator is measured to produce a beat frequency bias of 69 Hz, which agrees well with the theoretical expectation