We consider the behaviour of an Ising ferromagnet obeying the Glauber dynamics under the influence of a fast-switching, random external field. In paper I, we introduced a general formalism for describing such systems and presented the mean-field theory. In this article we derive results for the one-dimensional case, which can be only partially solved. Monte Carlo simulations performed on a square lattice indicate that the main features of the mean-field theory survive the presence of strong fluctuations.