Attitude control of a spacecraft using a variable-speed double-gimbal control moment gyro is considered in this paper. A steering law of this type of actuator has singular points where the rotation axis of the wheel coincides with that of the outer gimbal. Either the avoidance or passage of the singular states can be adopted in the steering law. In this paper, the steering law that selects the avoidance or passage of the singular states from the outer gimbal motion in the vicinity of the singular states is proposed. The usefulness of the steering law is examined by the ground experiments as well as the numerical simulations.