In a process for converting a heavy hydrocarbon fraction comprising conducting hydroconversion in a section for hydroconversion in the presence of hydrogen, the section comprising at least one three-phase reactor containing at least one ebullated bed of hydroconversion catalyst operating in riser mode for liquid and for gas, said reactor comprising at least one means for extracting used catalyst from said reactor and at least one means for adding fresh catalyst to said reactor, under conditions which produce a liquid feed with a reduced Conradson carbon, a reduced metal content, and a reduced sulphur content, the improvement wherein fresh catalyst is pre-conditioned before being injected into the reactor or reactors, said pre-conditioning producing a catalyst which is at least partially sulphurised, said fresh catalyst having an active phase comprising group VIII metals combined with group VIB metals on a support consisting essentially of a mineral oxide selected from the group consisting of alumina and silica.