The motion of a heavy rigid body with a single fixed point in a uniform gravity field is considered. The geometry of the mass of the body and the initial conditions of its motion correspond to the case of Goryachev-Chaplygin integrability [1,2]. In this case periodic pendulum-like motions exist, corresponding to oscillations or rotations of the body around an axis of dynamic symmetry, occupying a fixed horizontal position. The problem of the orbital stability of such motions is solved. An explicit solution of the linearized equations of the perturbed motion is obtained and it is shown that, in the linear approximation, the oscillations and rotations of the body are orbitally stable, while the non-linear problem of stability is always a resonance problem: for any amplitude of the oscillations (or any angular velocity of rotation) of the body in unperturbed motion its perturbed motion is such that fourthorder resonance occurs (two non-unity multipliers are pure imaginary and equal to _+i). It is shown that, in the non-linear formulation of the problem, the pendulum-like oscillations of the body are always orbitally unstable, while the rotations are stable.