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The lack of bright host galaxies in several recently examined gamma-ray burst(GRB) error boxes suggests that the redshifts of cosmological GRBs may besignificantly higher than previously believed. On the other hand, thenon-detection of multiple images in the BATSE 4B catalog implies an upper limiton the average redshift <z> of GRBs. Here, we calculate an upper limit to <z>,independent of the physical model for GRBs, using a new statistical lensingmethod that removes distance ambiguities, and thus permits accurate computationof the lensing rate at high z. The upper limit on <z> depends directly on thecosmological parameters Omega and Lambda. If there are no multiple images amongthe brightest 80% of the first 1802 bursts in the BATSE 4B catalog, then, atthe 95% confidence level, <z> is less than 2.2, 2.8, 4.3, or 5.3 for(Omega,Lambda) values of (0.3, 0.7), (0.5, 0.5), (0.5, 0.0), or (1.0, 0.0),respectively. The 68% upper limit to the average redshift is comparable to orless than the median redshift of GRBs in scenarios in which the GRB rate isproportional to the rate of star formation, for any cosmology. The uncertaintyin the lensing rate --- arising from uncertainties in the cosmologicalparameters and in the number density and average velocity dispersion ofgalaxies --- will be reduced significantly in the next few years by a newgeneration of experiments and surveys. Moreover, the continued increase in thenumber of GRBs observed by BATSE will greatly constrain their redshiftdistribution.Comment: 24 pages (LaTex, uses aaspp4.sty), with 4 encapsulated PostScript  figures, and 1 table. To appear in The Astrophysical Journal, Vol. 510,  January 1, 1999. Upper limits have been strengthened significantly because of  an increase in the BATSE 4B catalog estimated burst detection efficienc

Gravitational Lensing Limits on the Average Redshift of Gamma‐Ray Bursts