Data acquired with the COBE Differential Microwave Radiometer (DMR) provide a unique opportunity to observe simultaneous emission from cosmic gamma-ray bursts in the previously unexplored microwave region of the spectrum. We have searched the COBE DMR time-ordered data sets for instances when one of the DMR horns (FWHM ~ 7°) was pointing in the direction of a gamma-ray burst at the time of burst occurrence. During the overlap period 1991 April-December corresponding to the first public release of COBE data, 210 Compton Gamma Ray Observatory (CGRO)/BATSE gamma-ray bursts listed in the Third BATSE (3B) Catalog were viewable by the COBE DMR. For five of these events the DMR was pointing within 7° of the burst positions at the exact moment of burst occurrence. For another four events the DMR was pointed within 2° of the BATSE positions within 10 s of the burst trigger time. No obvious microwave emission (at 31.5, 53, or 90 GHz), with upper limits in the 10-100 kJy range, can be associated with any of these events. The COBE DMR has a relatively low sensitivity for the detection of point sources within its field of view. A positive detection of a gamma-ray burst by the COBE DMR would imply that the integrated microwave flux must be of the same order as the energy observed in gamma rays. By extending an acceptance window in time of up to 20 minutes before and after a gamma-ray burst another 60 bursts are sampled by the DMR, whose signals are analyzed statistically. We conclude that the "average" gamma-ray burst produces less than about 7-42 kJy in simultaneous microwave radiation.
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