  Gauge mediation of supersymmetry breakdown has many attractive features and
can be realized in phenomenologically interesting string-motivated models. We
point out that in models with the Polonyi-like field stabilized at a low
expectation value by quantum corrections, gravity seems to limit from above the
admixture of gravity mediation to the dominant gauge mediation channel.
However, we also point out that in a class of such models the low energy
metastable supersymmetry breaking vaccum appears to be cosmologically
disfavoured. These features should hold also in the case of typical stabilized
models with anomalous U(1) groups.
