A process for the hydrotreatment or hydroconversion of gas oils, vacuum distillates, atmospheric or vacuum residues, using a facility comprising at least: • a reaction section R-1, • a high pressure hot separator drum B-1, supplied with the effluent obtained from the reaction section R-1 and from which the bottom stream is supplied to the separator drum B-5, • a high pressure cold separator drum B-2, supplied with the overhead stream leaving the high pressure hot separator drum B-1 and from which the bottom stream is supplied to the stripper C-1, • a compression zone K for the gaseous effluent obtained from B-2, termed the recycled hydrogen, • a low pressure hot separator drum B-5, supplied with the liquid stream obtained from B-1, and from which the overhead gaseous effluent constitutes a portion of the feed for the stripper C-1, and from which the liquid effluent constitutes the first portion of the feed for the fractionation column C-2, • a separation column C-1 also termed a stripper supplied with the liquid stream obtained from B-2, and the gaseous stream obtained from B-5, from which the bottom product constitutes the other portion of the feed for the fractionation column C-2, • a principal fractionation column C-2, supplied with the bottom product from the stripper C-1 and with the liquid stream obtained from the bottom of B-5, and which separates the following cuts: naphtha light and heavy, diesel, kerosene and residue, • a furnace F-1 heating the feed for the reaction section R-1 and/or a portion of the hydrogen necessary for said reaction section, said separation column C-1 being operated under the following conditions: total pressure in the range 0.6 to 2.0 MPa, preferably in the range 0.7 to 1.8 MPa, and said fractionation column C-2 being operated under the following pressure conditions: total pressure in the range 0.1 MPa to 0.4 MPa, preferably in the range 0.1 MPa to 0.3 MPa. • a reaction section R-1, • a high pressure hot separator drum B-1, supplied with the effluent obtained from the reaction section R-1 and from which the bottom stream is supplied to the separator drum B-5, • a high pressure cold separator drum B-2, supplied with the overhead stream leaving the high pressure hot separator drum B-1 and from which the bottom stream is supplied to the stripper C-1, • a compression zone K for the gaseous effluent obtained from B-2, termed the recycled hydrogen, • a low pressure hot separator drum B-5, supplied with the liquid stream obtained from B-1, and from which the overhead gaseous effluent constitutes a portion of the feed for the stripper C-1, and from which the liquid effluent constitutes the first portion of the feed for the fractionation column C-2, • a separation column C-1 also termed a stripper supplied with the liquid stream obtained from B-2, and the gaseous stream obtained from B-5, from which the bottom product constitutes the other portion of the feed for the fractionation column C-2, • a principal fractionation column C-2, supplied with the bottom product from the stripper C-1 and with the liquid stream obtained from the bottom of B-5, and which separates the following cuts: naphtha light and heavy, diesel, kerosene and residue, • a furnace F-1 heating the feed for the reaction section R-1 and/or a portion of the hydrogen necessary for said reaction section,