Stellar-mass black holes have all been discovered through X-ray emission, which arises from the accretion of gas from their binary companions (this gas is either stripped from low-mass stars or supplied as winds from massive ones). Binary evolution models also predict the existence of black holes accreting from the equatorial envelope of rapidly spinning Be-type stars (stars of the Be type are hot blue irregular variables showing characteristic spectral emission lines of hydrogen). Of the approximately 80 Be X-ray binaries known in the Galaxy, however, only pulsating neutron stars have been found as companions. A black hole was formally allowed as a solution for the companion to the Be star MWC656 (ref. 5; also known as HD215227), although that conclusion was based on a single radial velocity curve of the Be star, a mistaken spectral classification and rough estimates of the inclination angle. Here we report observations of an accretion disk line mirroring the orbit of MWC656. This, together with an improved radial velocity curve of the Be star through fitting sharp FeII profiles from the equatorial disk, and a refined Be classification (to that of a B1.5-B2 III star), indicates that a black hole of 3.8 to 6.9 solar masses orbits MWC656, the candidate counterpart of the γ-ray source AGLJ2241+4454 (refs 5, 6). The black hole is X-ray quiescent and fed by a radiatively inefficient accretion flow giving a luminosity less than 1.6×10-7 times the Eddington luminosity. This implies that Be binaries with black-hole companions are difficult to detect in conventional X-ray surveys.