Recent observations and theoretical considerations have linked gamma-ray bursts with ultra-bright type Ibc supernovae (`hypernovae'). We here work out a specific scenario for this connection. Based on earlier work, we argue that especially the longest bursts must be powered by the Blandford-Znajek mechanism of electromagnetic extraction of spin energy from a black hole. Such a mechanism requires a high angular momentum in the progenitor object. The observed association of gamma-ray bursts with type Ibc supernovae leads us to consider massive helium stars that form black holes at the end of their lives as progenitors. In our analysis we combine the numerical work of MacFadyen & Woosley with analytic calculations in Kerr geometry, to show that about 10 53 erg each are available to drive the fast GRB ejecta and the supernova. The GRB ejecta are driven by the power output through the open field lines threading the black hole, whereas the supernova can be powered both by the shocks driven into the envelope by the jet, and by the power delivered into the disk via field lines connecting the disk with the black hole. We also present a much simplified approximate derivation of these energetics. Helium stars that leave massive black-hole remnants can only be made in fairly specific binary evolution scenarios, namely the kind that also leads to the formation of soft X-ray transients with black-hole primaries, or in very massive WNL stars. Since the binary progenitors will inevitably possess the high angular momentum we need, we propose a natural link between black-hole transients and gamma-ray bursts. Recent observations of one such transient, GRO J1655-40/Nova Scorpii 1994, explicitly support this connection: its high space velocity indicates that substantial mass was ejected in the formation of the black hole, and the overabundance of α-nuclei, especially sulphur, indicates that the explosion energy was extreme, as in SN 1998bw/GRB 980425. Furthermore, X-ray studies of this object indicate that the black hole may still be spinning quite rapidly, as expected in our model. We also show that the presence of a disk during the powering of the GRB and the explosion is required to deposit enough of the α nuclei on the companion.