Process for converting a heavy hydrocarbon feedstock containing a fraction of at least 50% having a boiling point of at least 300°C, and containing sulfur, Conradson carbon, metals and nitrogen, comprising the following successive steps: - an initial step of hydroconversion ( a 1) of at least a portion of said heavy hydrocarbon feedstock in the presence of hydrogen in an initial hydroconversion section ( A 1), performed under conditions for obtaining a liquid effluent with a reduced content of sulfur, Conradson carbon, metals and nitrogen; - (n-1) additional hydroconversion step(s) ( a i) in (n-1) additional hydroconversion section(s) ( A i), in the presence of hydrogen, of at least a portion or all of the liquid effluent obtained from the preceding hydroconversion step ( a i -1) or optionally of a heavy fraction obtained from an optional intermediate separation step ( b j) in an intermediate separation section ( B j) between two consecutive hydroconversion steps separating a portion or all of the liquid effluent obtained from the preceding hydroconversion step ( a i -1) to produce at least one heavy fraction predominantly boiling at a temperature of greater than or equal to 350°C, the (n-1) additional hydroconversion step(s) ( a i) being performed so as to obtain a hydroconverted liquid effluent with a reduced content of sulfur, Conradson carbon, metals and nitrogen, n being the total number of hydroconversion steps, with n greater than or equal to 2, i being an integer ranging from 2 to n and j being an integer ranging from 1 to (n-1), and the initial hydroconversion section ( A 1) and additional hydroconversion section ( A 1) each including at least one three-phase reactor operating as an ebullated bed or as a hybrid bed, containing at least one hydroconversion catalyst which is used in the form of extrudates or beads; - a first step of fractionation ( c) in a first fractionation section of a portion or all of the hydroconverted liquid effluent obtained from the last additional hydroconversion step (a n) producing at least one heavy cut predominantly boiling at a temperature of greater than or equal to 350°C, said heavy cut containing a residual fraction boiling at a temperature of greater than or equal to 540°C; - a step of deasphalting ( d) in a deasphalter of a portion or all of said heavy cut obtained from the fractionation step ( c), with at least one hydrocarbon-based solvent, to obtain a deasphalted oil DAO and a residual asphalt; - optionally a second step of fractionation ( e) in a second fractionation section of a portion or all of the DAO obtained from the deasphalting step ( d) into at least one heavy DAO fraction and one light DAO fraction; - a step of recycling ( f) of at least a portion of the DAO obtained from step ( d) and/or of at least a portion of the heavy DAO fraction obtained from step ( e) into an additional hydroconversion step (a i) and/or into an intermediate separation step (b j). - an initial step of hydroconversion ( a 1) of at least a portion of said heavy hydrocarbon feedstock in the presence of hydrogen in an initial hydroconversion section ( A 1), performed under conditions for obtaining a liquid effluent with a reduced content of sulfur, Conradson carbon, metals and nitrogen; - (n-1) additional hydroconversion step(s) ( a i) in (n-1) additional hydroconversion section(s) ( A i), in the presence of hydrogen, of at least a portion or all of the liquid effluent obtained from the preceding hydroconversion step ( a i -1) or optionally of a heavy fraction obtained from an optional intermediate separation step ( b j) in an intermediate separation section ( B j) between two consecutive hydroconversion steps separating a portion or all of the liquid effluent obtained from the preceding hydroconversion step ( a i -1) to produce at least one heavy fraction predominantly boiling at a temperature of greater than or equal to 350°C, the (n-1) additional hydroconversion step(s) ( a i) being performed so as to obtain a hydroconverted liquid effluent with a reduced content of sulfur, Conradson carbon, metals and nitrogen, - a first step of fractionation ( c) in a first fractionation section of a portion or all of the hydroconverted liquid effluent obtained from the last additional hydroconversion step (a n) producing at least one heavy cut predominantly boiling at a temperature of greater than or equal to 350°C, said heavy cut containing a residual fraction boiling at a temperature of greater than or equal to 540°C; - a step of deasphalting ( d) in a deasphalter of a portion or all of said heavy cut obtained from the fractionation step ( c), with at least one hydrocarbon-based solvent, to obtain a deasphalted oil DAO and a residual asphalt; - optionally a second step of fractionation ( e) in a second fractionation section of a portion or all of the DAO obtained from the deasphalting step ( d) into at least one heavy DAO fraction and one light DAO fraction; - a step of recycling ( f) of at least a portion of the DAO obtained from step ( d) and/or of at least a portion of the heavy DAO fraction obtained from step ( e) into an additional hydroconversion step (a i) and/or into an intermediate separation step (b j).