  We investigate the communication of supersymmetry breaking to the Standard
Model in theories of gauge mediated supersymmetry breaking with general weakly
coupled messenger sectors. We calculate the one loop gaugino and two loop soft
scalar masses for nonvanishing Str M^2 of the messenger sector. The soft scalar
masses are sensitive to physics at scales higher than the messenger scale, in
contrast to models with vanishing messenger supertrace. We discuss the
implications of this ultraviolet sensitivity in theories with renormalizable
and nonrenormalizable supersymmetry breaking sectors. We note that the standard
relation, in minimal gauge mediation, between soft scalar and gaugino masses is
altered in models with nonvanishing messenger supertrace.
