We report the first results of a programme to measure proper motions of stars in the innermost core of the Galaxy. From high-resolution near-infrared imaging over the last four years we have determined proper motions for 39 stars between 0.03 and 0.3 pc from the compact radio source Sgr A*. For 19 of these the derived motions are more significant than 4sigma in at least one coordinate. Proper motion and radial velocity dispersions are in very good agreement, indicating that the stellar velocity field on average is close to isotropic. Taking radial and proper motion data together the dynamic evidence is now strong that there is a 2.45(+/-0.4)x10^6- Msolar central dark mass located within <=0.015 pc of Sgr A*. Its mass density is at least 6.5x10^9 Msolar pc^-3, excluding the fact that the central mass concentration is in form of a compact white dwarf or neutron star cluster. In addition, we have detected significant changes in the structure of the innermost complex of stars in the immediate vicinity of Sgr A*, implying in at least one case stellar motions of >=1500 km s^-1 within ~0.01 pc of the compact radio source. Including this preliminary evidence, the inferred density of the central dark mass would then have to be in excess of 10^12 Msolar pc^-3, implying that the central mass concentration is probably a single massive black hole.