The processes that heat the solar and stellar coronae to several million kelvins, compared with the much cooler photosphere (5,800 K for the Sun), are still not well known 1. One proposed mechanism is heating via a large number of small, unresolved, impulsive heating events called nanoflares 2. Each event would heat and cool quickly, and the average effect would be a broad range of temperatures including a small amount of extremely hot plasma. However, detecting these faint, hot traces in the presence of brighter, cooler emission is observationally challenging. Here we present hard X-ray data from the second flight of the Focusing Optics X-ray Solar Imager (FOXSI-2), which detected emission above 7 keV from an active region of the Sun with no obvious individual X-ray flare emission. Through differential emission measure computations, we ascribe this emission to plasma heated above 10 MK, providing evidence for the existence of solar nanoflares. The quantitative evaluation of the hot plasma strongly constrains the coronal heating models.