  We discuss two cosmological issues in a generic gauge-mediated supersymmetry
(SUSY)-breaking model, namely the Universe's baryon asymmetry and the gravitino
dark-matter density. We show that both problems can be simultaneously solved if
there exist extra matter multiplets of a SUSY-invariant mass of the order of
the ``$\mu$-term'', as suggested in several realistic SUSY grand-unified
theories. We propose an attractive scenario in which the observed baryon
asymmetry is produced in a way totally independent of the reheating temperature
of inflation without causing any cosmological gravitino problem. Furthermore,
in a relatively wide parameter space, we can also explain the present mass
density of cold dark matter by the thermal relics of the gravitinos without an
adjustment of the reheating temperature of inflation. We point out that there
is an interesting relation between the baryon asymmetry and the dark-matter
density.
