  In this paper we study the nearest neighbor Ising model with ferromagnetic
interactions in the presence of a space dependent magnetic field which vanishes
as $|x|^{-\alpha}$, $\alpha >0$, as $|x|\to \infty$. We prove that in
dimensions $d\ge 2$ for all $\beta$ large enough if $\alpha>1$ there is a phase
transition while if $\alpha<1$ there is a unique DLR state.
