  The lack of bright host galaxies in several recently examined gamma-ray burst
(GRB) error boxes suggests that the redshifts of cosmological GRBs may be
significantly higher than previously believed. On the other hand, the
non-detection of multiple images in the BATSE 4B catalog implies an upper limit
on 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 lensing
method that removes distance ambiguities, and thus permits accurate computation
of the lensing rate at high z. The upper limit on <z> depends directly on the
cosmological parameters Omega and Lambda. If there are no multiple images among
the brightest 80% of the first 1802 bursts in the BATSE 4B catalog, then, at
the 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 or
less than the median redshift of GRBs in scenarios in which the GRB rate is
proportional to the rate of star formation, for any cosmology. The uncertainty
in the lensing rate --- arising from uncertainties in the cosmological
parameters and in the number density and average velocity dispersion of
galaxies --- will be reduced significantly in the next few years by a new
generation of experiments and surveys. Moreover, the continued increase in the
number of GRBs observed by BATSE will greatly constrain their redshift
distribution.
