A generic class of string theories predicts the existence of light moduli fields, and they are expected to have masses m_ϕ comparable to the gravitino mass m_3/2 which is in a range of 10^-2keV--1GeV in gauge-mediated supersymmetry breaking theories. Such light fields with weak interactions suppressed by the Planck scale can not avoid some stringent cosmological constraints, that is, they suffer from 'cosmological moduli problems'. We show that all the gravitino mass region 10^-2keV ≲ m_3/2≲

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... V is excluded by the constraints even if we incorporate a late-time mini-inflation (thermal inflation). However, a modification of the original thermal inflation model enables the region 10^-2keV ≲ m_3/2≲ 500keV to survive the constraints. It is also stressed that the moduli can be dark matter in our universe for the mass region 10^-2keV ≲ m_ϕ≲ 100keV.