The tight relationship between the masses of black holes and galaxy spheroids in nearby galaxies implies a causal connection between the growth of these two components. Optically luminous quasars host the most prodigious accreting black holes in the Universe, and can account for greater than or approximately equal to 30 per cent of the total cosmological black-hole growth. As typical quasars are not, however, undergoing intense star formation and already host massive black holes (> 10(8)M(o), where M(o) is the solar mass), there must have been an earlier pre-quasar phase when these black holes grew (mass range approximately (10(6)-10(8))M(o)). The likely signature of this earlier stage is simultaneous black-hole growth and star formation in distant (redshift z > 1; >8 billion light years away) luminous galaxies. Here we report ultra-deep X-ray observations of distant star-forming galaxies that are bright at submillimetre wavelengths. We find that the black holes in these galaxies are growing almost continuously throughout periods of intense star formation. This activity appears to be more tightly associated with these galaxies than any other coeval galaxy populations. We show that the black-hole growth from these galaxies is consistent with that expected for the pre-quasar phase.