Using balloon-borne instrumentation of very high sensitivity, approximately 25 hard X-ray bursts with peak fluxes of above 7 x 10 to the -3rd/(sq cm s keV) at 20 keV have been detected, in 141 minutes of observation of the sun on June 27, 1980. These hard X-ray microflares last from a few seconds to several tens of seconds and have power-law energy spectra. They are generally accompanied by small soft X-ray bursts, but H-alpha flares and solar radio bursts are reported for only a few of these hard X-ray bursts. The integral number of events varies approximately as the inverse of the peak flux, down to the limits of the measurements. These observations suggest that even very small transient releases of energy by the sun may be primarily nonthermal in character. It is speculated that the energy released in accelerated electrons for these microflares, averaged over time, may contribute significantly to the heating of the active corona.