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The distribution on the sky and the number-intensity distribution of gamma-ray bursts detected by BATSE indicate that the burst sources may be at cosmological distances, as suggested by Paczynski (Nature 355, 521, 1992). Bright BATSE bursts (peak intensities ${>}$ 18000 counts s$^{-1}$), which occupy the -3/2 power-law portion of the number-intensity relation, and dim bursts (1400--4500 counts s$^{-1}$), which lie in the regime where the relation is significantly flatter, were analyzed to search for a relative time-dilation effect. BATSE data in the 25--100 keV and 100--300 kev energy bands were used. Selection effects arising from intensity differences were eliminated by rescaling all burst temporal profiles to a canonical dim peak intensity and rendering their backgrounds and signal-to-noise levels equal. Pulse were then fitted to the major temporal structures in burst profiles, and distributions of pulse widths and intervals between pulses were generated for the fitted pulses. Gaussian fits, $\chi^2$ tests, and Kolmogorov-Smirnov tests were performed to search for and quantify any time-dilation signature that might be present. In the 25--100 keV band, a relative time-dilation factor is measured, between dim and bright bursts, of 2.5 $\pm$ 0.6 for widths and 2.2 $\pm$ 0.6 for intervals (90\% confidence levels), consistent with results from other tests on pulse-width and event-duration timescales (Norris et al. ApJ 424, 540, 1994; ApJ, in press, 1995). Because pulses tend to be narrower at higher energies, pulse-width time dilation would be expected to be almost completely masked in the 100--300 keV band, but not in the 25--100 keV band. In principle, pulse-interval time dilation would be unaffected by this spectral redshift. Results are consistent with this expectation: in the 100--300 keV band, a time-dilation factor of 2.2 $\pm$ 0.6 for pulse intervals is still manifest, while negligible time dilation for pulse widths is found.
