  The dilaton field in string theories (if exists) is expected to have a mass
of the order of the gravitino mass $m_{3/2}$ which is in a range of
$10^{-2}$keV--1GeV in gauge-mediated supersymmetry breaking models. If it is
the case, the cosmic energy density of coherent dilaton oscillation easily
exceeds the critical density of the present universe. We show that even if this
problem is solved by a late-time entropy production (thermal inflation) a
stringent constraint on the energy density of the dilaton oscillation is
derived from experimental upperbounds on the cosmic X($\gamma$)-ray
backgrounds. This excludes an interesting mass region, $500keV \lesssim m_{3/2}
\lesssim 1GeV$, in gauge-mediated supersymmetry breaking models.
