  We investigate nonequilibrium phase transitions in the presence of disorder
that locally breaks the symmetry between two equivalent macroscopic states. In
low-dimensional equilibrium systems, such "random-field" disorder is known to
have dramatic effects: It prevents spontaneous symmetry breaking and completely
destroys the phase transition. In contrast, we demonstrate that the phase
transition of the one-dimensional generalized contact process persists in the
presence of random field disorder. The dynamics in the symmetry-broken phase
becomes ultraslow and is described by a Sinai walk of the domain walls between
two different absorbing states. We discuss the generality and limitations of
our theory, and we illustrate our results by means of large-scale Monte-Carlo
simulations.
