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The dilaton field in string theories (if exists) is expected to have a massof 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 isthe case, the cosmic energy density of coherent dilaton oscillation easilyexceeds the critical density of the present universe. We show that even if thisproblem is solved by a late-time entropy production (thermal inflation) astringent constraint on the energy density of the dilaton oscillation isderived from experimental upperbounds on the cosmic X($\gamma$)-raybackgrounds. This excludes an interesting mass region, $500keV \lesssim m_{3/2}\lesssim 1GeV$, in gauge-mediated supersymmetry breaking models.Comment: 13 pages (RevTex file including one figure, use psfig), revised  version to be published in Physical Review Letter

Cosmological Constraint on the String Dilaton in Gauge-Mediated Supersymmetry Breaking Theories