The well-known central configurations of the three-body problem give rise to periodic solutions where the bodies rotate rigidly around their center of mass. For these solutions, the moment of inertia of the bodies with respect to the center of mass is clearly constant. Saari conjectured that such rigid motions, called relative equilibrium solutions, are the only solutions with constant moment of inertia. This result will be proved here for the Newtonian three-body problem in $\R^d$ with three positive masses. The proof makes use of some computational algebra and geometry. When $d\le 3$, the rigid motions are the planar, periodic solutions arising from the five central configurations, but for $d\ge 4$ there are other possibilities.