Process for converting a heavy feedstock of hydrocarbons having an initial boiling point of at least 300°C, a Conradson carbon content of greater than 5% by weight and a sulfur content of greater than 0.5% by weight, comprising the following steps: a) at least one step of selective deasphalting of the heavy feedstock of hydrocarbons by liquid/liquid extraction that makes it possible to separate at least one asphalt fraction and at least one deasphalted oil fraction, at least one of said deasphalting steps being performed by means of a mixture of at least one polar solvent and at least one nonpolar solvent, the proportion of said polar solvent in said mixture of polar solvent and nonpolar solvent being adjusted according to the properties of the treated feedstock and according to the required asphalt yield and/or required quality of the deasphalted oil, and being between 0.1% and 99.9% volume/volume, said polar solvent being selected from pure aromatic solvents or naphthene aromatics, polar solvents containing heteroelements, or mixtures thereof, said nonpolar solvent comprising a saturated hydrocarbon solvent containing a number of carbon atoms greater than or equal to 2, said deasphalting steps being carried out under conditions that are subcritical for the solvent mixture used, at a temperature lower than the critical temperature of the solvent mixture, b) a step of hydroconverting of the deasphalted oil fraction 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 having upflow of liquid and of gas and comprising at least one means of withdrawing said catalyst from said reactor and at least one means of contributing fresh catalyst to said reactor, under conditions that make it possible to obtain an outflow comprising a gaseous fraction containing mainly the compounds H 2 and H 2 S and a liquid fraction having a reduced Conradson carbon content and reduced content of metals, of sulfur and of nitrogen, c) a step of separating the outflow resulting from step b) to obtain a gaseous fraction containing mainly the compounds H 2 and H 2 S and a liquid fraction having a reduced Conradson carbon content and reduced content of metals, of sulfur and of nitrogen. a) at least one step of selective deasphalting of the heavy feedstock of hydrocarbons by liquid/liquid extraction that makes it possible to separate at least one asphalt fraction and at least one deasphalted oil fraction, at least one of said deasphalting steps being performed by means of a mixture of at least one polar solvent and at least one nonpolar solvent, the proportion of said polar solvent in said mixture of polar solvent and nonpolar solvent being adjusted according to the properties of the treated feedstock and according to the required asphalt yield and/or required quality of the deasphalted oil, and being between 0.1% and 99.9% volume/volume, said polar solvent being selected from pure aromatic solvents or naphthene aromatics, polar solvents containing heteroelements, or mixtures thereof, said nonpolar solvent comprising a saturated hydrocarbon solvent containing a number of carbon atoms greater than or equal to 2, said deasphalting steps being carried out under conditions that are subcritical for the solvent mixture used, at a temperature lower than the critical temperature of the solvent mixture, b) a step of hydroconverting of the deasphalted oil fraction 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 having upflow of liquid and of gas and comprising at least one means of withdrawing said catalyst from said reactor and at least one means of contributing fresh catalyst to said reactor, under conditions that make it possible to obtain an outflow comprising a gaseous fraction containing mainly the compounds H 2 and H 2 S and a liquid fraction having a reduced Conradson carbon content and reduced content of metals, of sulfur and of nitrogen, c) a step of separating the outflow resulting from step b) to obtain a gaseous fraction containing mainly the compounds H 2 and H 2 S and a liquid fraction having a reduced Conradson carbon content and reduced content of metals, of sulfur and of nitrogen.