EVIDENCE for a massive black hole at the centre of our Galaxy has been accumulating for the past two decades 1-7. Estimates of the mass of this region have hitherto been based on the spectroscopic determination of radial velocities for stars and gas near the Galactic Centre, combined with the assumption that the stars are moving on largely circular, isotropically distributed orbits. But if this assumption is incorrect, the observations can be explained with a much smaller central mass, perhaps obviating the need for a massive black hole. Here we report the proper motions (motions in the plane of the sky) of 39 stars located between 0.04 and 0.4 pc from the Galactic Centre. We find that the velocity dispersion estimated from the proper motions is in excellent agreement with that obtained from the radial velocities, indicating that the average velocity field is close to isotropic. Taken together, the observations provide strong evidence for a central dark mass of 2.45 +/- 0.4 × 106 solar masses (Msolar) located within 0.015 pc of the compact radio source Sgr A*. We place a lower limit of 6.5 × 109 Msolarpc-3 on the density of the central region, suggesting that it is most probably occupied by a massive black hole.