  We study the nonequilibrium phase transition in the two-dimensional contact
process on a randomly diluted lattice by means of large-scale Monte-Carlo
simulations for times up to $10^{10}$ and system sizes up to $8000 \times 8000$
sites. Our data provide strong evidence for the transition being controlled by
an exotic infinite-randomness critical point with activated (exponential)
dynamical scaling. We calculate the critical exponents of the transition and
find them to be universal, i.e., independent of disorder strength. The
Griffiths region between the clean and the dirty critical points exhibits
power-law dynamical scaling 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. Our results are of
importance beyond absorbing state transitions because according to a
strong-disorder renormalization group analysis, our transition belongs to the
universality class of the two-dimensional random transverse-field Ising model.
