  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. We measured the proper motions of more
than 6000 stars within ~1.0 pc of the supermassive black hole Sgr 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) x 10^6 solar masses. Although this
value is slightly lower than the current canonical value of 4.0x10^6 solar
masses, 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 0.5-1.5x10^6 solar masses 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.
