A process for converting a heavy hydrocarbon fraction comprises treating the hydrocarbon feed in a hydrodemetallization section, the section comprising at least one fixed bed hydrodemetallization catalyst. At least a portion of the hydrotreated liquid effluent from step a) is sent to an atmospheric distillation zone from which a distillate and an atmospheric residue are recovered; at least a portion of the atmospheric residue is sent to a vacuum distillation zone from which a vacuum distillate and a vacuum residue are recovered; at least a portion of the vacuum residue is sent to a deasphalting section from which a deasphalted hydrocarbon cut and residual asphalt are recovered; and at least a portion of the deasphalted hydrocarbon cut is sent to a hydrotreatment section from which a gas fraction, a fuel fraction and a heavier liquid fraction of the hydrotreated feed are recovered, said section comprising at least one three-phase reactor containing at least one ebullated bed hydrotreatment catalyst operating in liquid and gas riser mode, the reactor comprising at least one means for removing catalyst from the reactor and at least one means for adding fresh catalyst to the reactor.