A search for weak γ-ray bursts was conducted using a sensitive balloon-borne detector. One burst was detected in 64 hr of observation. The upper limit to the burst rate is 2300 bursts yr-1 above 6×10-7ergs cm-2 for simple spatial distribution models. Comparison with satellite results indicates that the slope of the log N-log S curve can be no steeper than -1 between 10-4 and 10-6ergs cm-2. A detailed procedure for calculating detector sensitivity to bursts is provided.