The small signal current sensitivity of twisted fibers is analyzed and calculated theoretically, and a new and more convenient method to measure the Verdet constant is proposed and verified experimentally. The results show that if the ratio of fiber twist rate and intrinsic linear birefringence is large than certain figure the value of measured Verdet constant may quite close to the theoretical calculation. This method provides a new approch to measure the Verdet constant of conventional single mode optical fibers.