RECENT observation's from the BATSE experiment on the Gamma Ray Observatory 1,2 demonstrate that the number of weak gamma-ray bursts is smaller than expected for a uniform distribution of source distances, and that the distribution of weak bursts is isotropic. This suggests that the sources are at cosmological, rather than intragalactic, distances 3,4. A number of possible tests for estimating the distances to gamma-ray bursters have been discussed 3, but all are difficult to implement in practice. There is therefore no further evidence at present to support a cosmological distance to the bursters. Here I describe a test of distances that might be done with the BATSE data, which is equivalent to the formulation of the Hubble diagram for galactic redshifts. If the typical redshift of gamma-ray bursters is of the order of unity, the weakest bursts should have softer spectra than the strongest. There is some evidence for such a correlation in the BATSE data 5. If confirmed, this would provide the first spectroscopic indication that the burst sources lie at cosmological distances. There is also evidence that the redshift effect has been detected in the duration of a subset of the BATSE bursts 6. If confirmed, this would be evidence that not only the photons but also the light curves are stretched by the cosmological redshift.
