A new generally covariant method to calculate the motion of the axis of an orbiting gyroscope, with respect to the direction of fixed stars, is presented. The method includes general-relativistic aberration effects and gives frame-independent results which can be compared directly with the proposed NASA gyroscope experiment. Quantitative predictions for circular motion in the Schwarzschild field agree with presently accepted values. The method is valid for more general cases and may be used to study other metrics and motions. (AIP)