We report on an observation of AU Mic taken with the Goddard High Resolution Spectrograph (GHRS) aboard the Hubble Space Telescope. The data consist of a rapid sequence of spectra covering the wavelength range 1345-1375 A with a spectral resolution of 10,000. The observations were originally intended to search for spectral variations during flares. No flares were detected during the 3.5 hr of monitoring. A method of reducing the noise while combining the individual spectra in the time series is described which resulted in the elimination of half of the noise while rejecting only a small fraction of the stellar signal. The resultant spectrum was of sufficient quality to allow the detection of emission lines with an integrated flux of 10(exp -15) ergs/sq cm(sec) or greater. Lines of C I, O I, O V, Cl I, and Fe XXI were detected. This is the first indisputable detection of the 1354 A Fe XXI line, formed at T approximately = 10(exp 7) K, on a star other than the Sun. The line was well resolved and displayed no significant bulk motions or profile asymmetry. From the upper limit on the observed line width, we derive an upper limit of 38 km/s for the turbulent velocity in the 10(exp 7) K plasma. An upper limit is derived for the flux of the 1349 A Fe XII line, formed at T approximately = 1.3 x 10(exp 6) K. These data are combined with contemporaneous GHRS and International Ultraviolet Explorer (IUE) data to derive the volume emission measure distribution of AU Mic over the temperature range 10(exp 4)-10(exp 7) K. Models of coronal loops in hydrostatic equilibrium are consistent with the observed volume emission measures of the coronal lines. The fraction of the stellar surface covered by the footprints of the loops depends upon the loop length and is less than 14% for lengths smaller than the stellar radius. From the upper limit to the estimated width of the Fe XXI line prfile we find that the we cannot rule out Alfven wave dissipation as a possible contributor to the required quiescent loop heating rate.