  We explore the implications of a 124-126 GeV CP-even Higgs boson on the
fundamental parameter space and sparticle spectroscopy of the minimal gauge
mediated supersymmetry breaking (mGMSB) scenario. The above mass for the Higgs
boson yields stringent lower bounds on the sparticle masses in this class of
models. The lightest neutralino and stau masses lie close to 1.5 TeV and 800
GeV respectively, while the majority of the sparticle masses are in the several
to multi-TeV range. We show that with a single pair of 5+\bar{5} SU(5)
messenger multiplets, the lower limit on the gravitino mass is \sim 360 eV.
This is reduced to about 60 eV if five pairs of 5+\bar{5} messenger fields are
introduced. Non-standard cosmology and non-standard gravitino production
mechanisms are required in order to satisfy cosmological observations.
