  We study soft leptogenesis in gauge mediated supersymmetry breaking models
with an enhanced A-term for the right-handed neutrino. We find that this
scenario can explain the baryon asymmetry of the present universe, consistently
with the gravitino dark matter for a wide range of gravitino mass
m_{3/2}=O(MeV)-O(GeV). We also propose an explicit model which induces the
necessary A-term for the right-handed neutrino.
