This paper addresses the problem of stabilization of a monoaxial gyrostabilizer under the influence of uncertain external disturbances acting with respect to the stabilization axis, based on noisy measurements of the angle position of the preces- sion axis. The stabilization system is realized as a digital feedback linear dynamic controller designed with allowance for correlated measurement noise with unknown distribution parameters (with uncertain covariance matrices) in the measure- ment channel. To solve the problem of synthesis of the stabilizing control law, linear matrix inequalities and convex optimization are used. The designed control system is compared (by means of simulation) with some stabilizing controllers computed by other methods.