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\noindent The majority of gamma-ray bursts exhibit a peak in their $\nu{\cal F}_{\nu}$ photon energy spectra at an energy $E_p$ that is in the range of $\sim$20--2000 keV available to the Large Area Detectors of the Burst and Transient Source Experiment (BATSE) on the Compton Gamma--Ray Observatory. If gamma--ray burst sources are at cosmological distances ($d$ $\sim$1 Gpc), then the spectra of dim bursts should be redshifted to lower energies relative to those of bright bursts. The magnitude of the shift is a function of the cosmological redshifts $z$ of both the dim and bright burst sources and hence yields the range of redshift available to the bursts; this range is further constrained by considering cosmological model fits to the differential number--intensity distribution. We have produced photon energy spectra for $\sim$400 bursts using data from BATSE to investigate if this expected shift in the $\nu{\cal F}_{\nu}$ peak is observed. We find that the mean peak energies of the burst spectra are correlated with intensity: lower intensity groups of burst spectra exhibit a lower average peak energy, although the distributions of $E_p$ are quite wide. Denoting the redshift of an event with an observed photon flux $P$ (photons cm$^{-2}$ s$^{-1}$) by $z_P$, we find that the maximum range consistent with the bursts of our sample is $(1 + z_{1}) / (1 + z_{100}) \sim$2.2.
