  A generic class of string theories predicts the existence of light moduli
fields, and they are expected to have masses $m_\phi$ comparable to the
gravitino mass $m_{3/2}$ which is in a range of $10^{-2}$keV--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 $10^{-2}$keV $\lesssim
m_{3/2} \lesssim$ 1GeV 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 $10^{-2}$keV $\lesssim
m_{3/2} \lesssim$ 500keV to survive the constraints. It is also stressed that
the moduli can be dark matter in our universe for the mass region $10^{-2}$keV
$\lesssim m_\phi \lesssim$ 100keV.
