The origin of the hard (2-10 keV) X-ray background has been a mystery for over 35 years. Most of the soft X-ray background has been resolved into individual sources (mainly quasars), but these sources do not have the spectral energy distribution required to match the spectrum of the X-ray background as a whole. Here we report the results of a deep survey, using the Chandra satellite, in which the detected hard X-ray sources account for at least 75 per cent of the hard X-ray background. The mean X-ray spectral energy distribution of these sources is in good agreement with that of the background. Moreover, most of those hard X-ray sources are associated unambiguously with either the nuclei of otherwise normal bright galaxies, or with optically faint sources. The latter could be active nuclei in dust-enshrouded galaxies or a population of quasars at extremely high redshift.