  We present a new three-dimensional fully general-relativistic hydrodynamics
code using high-resolution shock-capturing techniques and a conformal traceless
formulation of the Einstein equations. Besides presenting a thorough set of
tests which the code passes with very high accuracy, we discuss its application
to the study of the gravitational collapse of uniformly rotating neutron stars
to Kerr black holes. The initial stellar models are modelled as relativistic
polytropes which are either secularly or dynamically unstable and with angular
velocities which range from slow rotation to the mass-shedding limit. We
investigate the gravitational collapse by carefully studying not only the
dynamics of the matter, but also that of the trapped surfaces, i.e. of both the
apparent and event horizons formed during the collapse. The use of these
surfaces, together with the dynamical horizon framework, allows for a precise
measurement of the black-hole mass and spin. The ability to successfully
perform these simulations for sufficiently long times relies on excising a
region of the computational domain which includes the singularity and is within
the apparent horizon. The dynamics of the collapsing matter is strongly
influenced by the initial amount of angular momentum in the progenitor star
and, for initial models with sufficiently high angular velocities, the collapse
can lead to the formation of an unstable disc in differential rotation. All the
simulations performed with uniformly rotating initial data and a polytropic or
ideal-fluid equation of state show no evidence of shocks or of the presence of
matter on stable orbits outside the black hole.
