The isotropic distribution of gamma-ray bursts as observed with the Burst and Transient Experiment (BATSE) strongly suggests that the bursts are at cosmological distances. At such distances, the expansion of the universe should redshift the spectra and stretch the temporal structure. We relate the observed peak intensities, spectral shapes, and time dilation to distance. We include the uncertainties in our knowledge of the intrinsic spectrum and correct for the coupling between the spectral shape and the temporal structure. The reported time dilation between the dimmest BATSE bursts and the bright BATSE bursts (a factor of 2) requires a redshift of about 6 for the BATSE dimmest bursts rather than a redshift of 1 or 2 as previously reported. The large luminosity and distance determined from the time dilation is inconsistent with the observed Log N-Log P distribution. unless there is very strong evolution. The implied distance for the dimmest bursts is beyond where galaxies are thought to form. If true, the gamma-ray bursts would be orphans: no known objects would have the same distance scale. We conclude that either a large fraction (65%) of the observed time dilation between the BATSE bright and dimmest bursts is intrinsic to the bursts or that there are strong evolutionary effects in the Log N-Log P distribution and that it is only a coincidence that Log N-Log P shows a -3/2 power law at high intensities.