Process for the conversion of a heavy hydrocarbon feedstock having an initial boiling point of at least 300°C comprising the following stages: a) a stage of hydroconversion of at least a part of said feedstock in the presence of hydrogen in at least one three-phase reactor, said reactor containing at least one hydroconversion catalyst and operating as an ebullating bed, with ascending flow of liquid and of gas and comprising at least one means for drawing off said catalyst from said reactor and at least one means for supplying fresh catalyst to said reactor, under conditions making it possible to obtain a liquid feedstock with a reduced content of Conradson carbon, metals, sulphur and nitrogen, b) a stage of separation of the effluent originating from stage a) in order to obtain a light liquid fraction with a boiling point lower than 300°C and a heavy liquid fraction with a boiling point greater than 300°C, c) at least two stages of deasphalting in series on at least a part of the heavy liquid fraction originating from stage b), making it possible to separate at least one fraction of asphalt, at least one fraction of heavy deasphalted oil, referred to as heavy DAO and at least one fraction of light deasphalted oil referred to as light DAO, at least one of said stages of deasphalting being carried out by means of a mixture of at least one polar solvent and at least one apolar solvent, the proportions of said polar solvent and said apolar solvent in the mixture of solvents being adjusted depending on the properties of the feedstock treated and depending on the desired yield of asphalt and/or quality of the deasphalted oil, said stages of deasphalting being implemented under the subcritical conditions of the mixture of solvents used, d) a stage of recycling at least a part of said heavy deasphalted oil cut referred to as heavy DAO originating from stage c) upstream of the stage a) of hydroconversion. a) a stage of hydroconversion of at least a part of said feedstock in the presence of hydrogen in at least one three-phase reactor, said reactor containing at least one hydroconversion catalyst and operating as an ebullating bed, with ascending flow of liquid and of gas and comprising at least one means for drawing off said catalyst from said reactor and at least one means for supplying fresh catalyst to said reactor, under conditions making it possible to obtain a liquid feedstock with a reduced content of Conradson carbon, metals, sulphur and nitrogen, b) a stage of separation of the effluent originating from stage a) in order to obtain a light liquid fraction with a boiling point lower than 300°C and a heavy liquid fraction with a boiling point greater than 300°C, c) at least two stages of deasphalting in series on at least a part of the heavy liquid fraction originating from stage b), making it possible to separate at least one fraction of asphalt, at least one fraction of heavy deasphalted oil, referred to as heavy DAO and at least one fraction of light deasphalted oil referred to as light DAO, at least one of said stages of deasphalting being carried out by means of a mixture of at least one polar solvent and at least one apolar solvent, the proportions of said polar solvent and said apolar solvent in the mixture of solvents being adjusted depending on the properties of the feedstock treated and depending on the desired yield of asphalt and/or quality of the deasphalted oil, said stages of deasphalting being implemented under the subcritical conditions of the mixture of solvents used, d) a stage of recycling at least a part of said heavy deasphalted oil cut referred to as heavy DAO originating from stage c) upstream of the stage a) of hydroconversion.