  Some gamma-ray burst (GRB) spectra exhibit high energy tails with the highest
photon energy detected at 18 GeV. The spectral slope of the high-energy tails
is sufficiently flat in nu F_nu to consider the possibility of their detection
at still higher energies. We calculate how many bursts can reasonably be
expected above a given energy threshold for a cosmological distribution of
bursts satisfying the observed apparent brightness distribution. The crucial
point is that the gamma-ray absorption by pair production in the intergalactic
diffuse radiation field eliminates bursts from beyond the gamma-ray horizon tau
~ 1, thus drastically reducing the number of bursts at high energies. Our
results are consistent with the non-detection of bursts by current experiments
in the 100 GeV to 100 TeV energy range. For the earth-bound detector array
MILAGRO, we predict a maximal GRB rate of ~ 10 events per year. The Whipple
Observatory can detect, under favorable conditions, ~1 event per year. The
event rate for the HEGRA array is ~ 0.01 per year. Detection of significantly
higher rates of bursts would severely challenge cosmological burst scenarios.
