Process for converting a heavy hydrocarbon feedstock with initial boiling point of at least 300° C. comprising: a) hydroconverting at least part of said feedstock; b) separating the effluent from stage a) to obtain light and heavy liquid fractions; c) at least two stages of deasphalting in series at least part of the heavy liquid fraction originating from stage b), allowing to separate at least one fraction of asphalt, at least one fraction of heavy deasphalted oil (heavy DAO) and at least one fraction of light deasphalted oil (light DAO), at least one deasphalting stage being carried out using a mixture of at least one polar solvent and at least one apolar solvent, said deasphalting stages being under subcritical conditions of the mixture of solvents; d) recycling at least part of said heavy deasphalted oil cut from stage c) upstream of hydroconverting a) and/or of the inlet of separating b).