Nuclear star clusters (NSCs) are located at the photometric and dynamical centers of the majority of galaxies. They are among the densest star clusters in the Universe. The NSC in the Milky Way is the only object of this class that can be resolved into individual stars. The massive black hole Sagittarius A* is located at the dynamical center of the Milky Way NSC. In this work we examine the proper motions of stars out to distances of 1.0 pc from Sgr A*. The aim is to examine the velocity structure of the MW NSC and acquire a reliable estimate of the stellar mass in the central parsec of the MW NSC, in addition to the well-known black hole mass. We use multi-epoch adaptive optics assisted near-infrared observations of the central parsec of the Galaxy obtained with NACO/CONICA at the ESO VLT. Stellar positions are measured via PSF fitting in the individual images and transformed into a common reference frame via suitable sets of reference stars. We measured the proper motions of more than 6000 stars within 1.0 pc of Sagittarius A*. The full data set is provided in this work. We largely exclude the known early-type stars with their peculiar dynamical properties from the dynamical analysis. The cluster is found to rotate parallel to Galactic rotation, while the velocity dispersion appears isotropic (or anisotropy may be masked by the cluster rotation). The Keplerian fall-off of the velocity dispersion due to the point mass of Sgr A* is clearly detectable only at R<= 0.3 pc. Nonparametric isotropic and anisotropic Jeans models are applied to the data. They imply a best-fit black hole mass of 3.6+0.2_-0.4*106 M_o. Although this value is slightly lower than the current canonical value of 4.0*106 M_o, this is the first time that a proper motion analysis provides a mass for Sagittarius A* that is consistent with the mass inferred from orbits of individual stars. The point mass of Sagittarius A* is not sufficient to explain the velocity data. In addition to the black hole, the models require the presence of an extended mass of 0.5-1.5* 106 M_o in the central parsec. This is the first time that the extended mass of the nuclear star cluster is unambiguously detected. The influence of the extended mass on the gravitational potential becomes notable at distances >= 0.4 pc from Sgr A*. Constraints on the distribution of this extended mass are weak. The extended mass can be explained well by the mass of the stars that make up the cluster.