The slope of a gamma-ray burst spectrum determines how the burst flux decreases as the cosmological red-shift z increases. This provides a test for the presence of a cosmological redshift in a sample of gamma-ray bursts. Many bursts have spectra characterized by a frequency-dependent power-law index, so a burst subset can be created by requiring that each subset member have a power-law index of the given value at a low frequency and an index of a different given value at a high frequency. If all bursts in this subset are at z much less than 0.1, then the flux at low frequency will change proportionally with the flux at high frequency. If, however, z > 0.1 for most bursts, then the linear dependence of the high-frequency flux on the low-frequency flux disappears at the flux characterizing the spatial cutoff in the log N-log C(max)/C(min) diagram. This signature of the cosmological redshift generally persists when gamma-ray bursts are given a broad distribution of intrinsic fluxes and spectral shapes. The characteristics that a burst distribution must have to mask the cosmological redshift are discussed. The limitations of this test are observational, arising primarily from errors in measuring the photon flux and in determining the spectral index of the burst spectrum.
