Theoretical models for the production of relativistic jets from active galactic nuclei predict that jet power arises from the spin and mass of the central supermassive black hole, as well as from the magnetic field near the event horizon. The physical mechanism underlying the contribution from the magnetic field is the torque exerted on the rotating black hole by the field amplified by the accreting material. If the squared magnetic field is proportional to the accretion rate, then there will be a correlation between jet power and accretion luminosity. There is evidence for such a correlation, but inadequate knowledge of the accretion luminosity of the limited and inhomogeneous samples used prevented a firm conclusion. Here we report an analysis of archival observations of a sample of blazars (quasars whose jets point towards Earth) that overcomes previous limitations. We find a clear correlation between jet power, as measured through the {gamma}-ray luminosity, and accretion luminosity, as measured by the broad emission lines, with the jet power dominating the disk luminosity, in agreement with numerical simulations9. This implies that the magnetic field threading the black hole horizon reaches the maximum value sustainable by the accreting matter.