  We study the nonequilibrium phase transition in the one-dimensional contact
process with quenched spatial disorder by means of large-scale Monte-Carlo
simulations for times up to $10^9$ and system sizes up to $10^7$ sites. In
agreement with recent predictions of an infinite-randomness fixed point, our
simulations demonstrate activated (exponential) dynamical scaling at the
critical point. The critical behavior turns out to be universal, even for weak
disorder. However, the approach to this asymptotic behavior is extremely slow,
with crossover times of the order of $10^4$ or larger. In the Griffiths region
between the clean and the dirty critical points, we find power-law dynamical
behavior with continuously varying exponents. We discuss the generality of our
findings and relate them to a broader theory of rare region effects at phase
transitions with quenched disorder.
