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Using data from the CYGNUS extensive air shower array, we have searched for evidence of emission of ultra-high-energy radiation coincident with $\gamma$-ray bursts observed by the BATSE instrument on the Compton Gamma-Ray Observatory. No statistically significant excess was found for any point in the sky within 4$\sigma$ of BATSE's best location coordinates for any of the 56 bursts examined. Furthermore, no events were seen in the 1.5$^{\circ}$ radius circular bin surrounding $\gamma$-ray burst GRB 920720, whose location was determined accurately by the Compton/Ulysses/PVO Interplanetary Network of satellites. Flux upper limits depend greatly on the actual zenith angle of the burst. Typical fluence upper limits above 100 TeV are $\sim$10$^{-6}$ erg cm$^{-2}$. The fluence upper limit for GRB 920720 is 2 $\times$ 10$^{-6}$ erg cm$^{-2}$.