The formation of the first massive objects in the infant Universe remains impossible to observe directly and yet it sets the stage for the subsequent evolution of galaxies(1-3). Although some black holes with masses more than 10(9) times that of the Sun have been detected in luminous quasars less than one billion years after the Big Bang(4,5), these individual extreme objects have limited utility in constraining the channels of formation of the earliest black holes; this is because the initial conditions of black hole seed properties are quickly erased during the growth process(6). Here we report a measurement of the amount of black hole growth in galaxies at redshift z = 6-8 (0.95-0.7 billion years after the Big Bang), based on optimally stacked, archival X-ray observations. Our results imply that black holes grow in tandem with their host galaxies throughout cosmic history, starting from the earliest times. We find that most copiously accreting black holes at these epochs are buried in significant amounts of gas and dust that absorb most radiation except for the highest-energy X-rays. This suggests that black holes grew significantly more during these early bursts than was previously thought, but because of the obscuration of their ultraviolet emission they did not contribute to the re-ionization of the Universe.