  We present a numerical code designed to study astrophysical phenomena
involving dynamical spacetimes containing black holes in the presence of
relativistic hydrodynamic matter. We present evolutions of the collapse of a
fluid star from the onset of collapse to the settling of the resulting black
hole to a final stationary state. In order to evolve stably after the black
hole forms, we excise a region inside the hole before a singularity is
encountered. This excision region is introduced after the appearance of an
apparent horizon, but while a significant amount of matter remains outside the
hole. We test our code by evolving accurately a vacuum Schwarzschild black
hole, a relativistic Bondi accretion flow onto a black hole, Oppenheimer-Snyder
dust collapse, and the collapse of nonrotating and rotating stars. These
systems are tracked reliably for hundreds of M following excision, where M is
the mass of the black hole. We perform these tests both in axisymmetry and in
full 3+1 dimensions. We then apply our code to study the effect of the stellar
spin parameter J/M^2 on the final outcome of gravitational collapse of rapidly
rotating n = 1 polytropes. We find that a black hole forms only if J/M^2<1, in
agreement with previous simulations. When J/M^2>1, the collapsing star forms a
torus which fragments into nonaxisymmetric clumps, capable of generating
appreciable ``splash'' gravitational radiation.
