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A generic class of string theories predicts the existence of light modulifields, and they are expected to have masses $m_\phi$ comparable to thegravitino mass $m_{3/2}$ which is in a range of $10^{-2}$keV--1GeV ingauge-mediated supersymmetry breaking theories. Such light fields with weakinteractions suppressed by the Planck scale can not avoid some stringentcosmological constraints, that is, they suffer from `cosmological moduliproblems'. We show that all the gravitino mass region $10^{-2}$keV $\lesssimm_{3/2} \lesssim$ 1GeV is excluded by the constraints even if we incorporate alate-time mini-inflation (thermal inflation). However, a modification of theoriginal thermal inflation model enables the region $10^{-2}$keV $\lesssimm_{3/2} \lesssim$ 500keV to survive the constraints. It is also stressed thatthe moduli can be dark matter in our universe for the mass region $10^{-2}$keV$\lesssim m_\phi \lesssim$ 100keV.Comment: A few changes in section IV and

Cosmological moduli problem in gauge-mediated supersymmetry breaking theories