We report the detection of emission from a nonthermal electron distribution in a small solar microflare (GOES class A5.7) observed by the Nuclear Spectroscopic Telescope Array, with supporting observation by the Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI). The flaring plasma is well accounted for by a thick-target model of accelerated electrons collisionally thermalizing within the loop, akin to the "coronal thick-target" behavior occasionally observed in larger flares. This is the first positive detection of nonthermal hard X-rays from the Sun using a direct imager (as opposed to indirectly imaging instruments). The accelerated electron distribution has a spectral index of 6.3 ± 0.7, extends down to at least 6.5 keV, and deposits energy at a rate of 2 × 1027 erg s-1, heating the flare loop to at least 10 MK. The existence of dominant nonthermal emission in X-rays down to <5 keV means that RHESSI emission is almost entirely nonthermal, contrary to what is usually assumed in RHESSI spectroscopy. The ratio of nonthermal to thermal energies is similar to that of large flares, in contrast to what has been found in previous studies of small RHESSI flares. We suggest that a coronal thick target may be a common property of many small microflares based on the average electron energy and collisional mean free path. Future observations of this kind will enable understanding of how flare particle acceleration changes across energy scales, and will aid the push toward the observational regime of nanoflares, which are a possible source of significant coronal heating.