The invention concerns a process for refining a heavy hydrocarbon feed, comprising the following steps:a) a step for selective deasphalting of the heavy hydrocarbon feed by single-step liquid/liquid extraction in an extraction medium, said extraction being carried out using a mixture of at least one polar solvent and at least one apolar solvent, in order to obtain an asphalt phase and a deasphalted oil phase DAO, the proportions of said polar solvent and said apolar solvent in the solvent mixture being adjusted as a function of the properties of the feed and the desired asphalt yield, said deasphalting step being carried out under subcritical conditions for the solvent mixture; b) a step for hydrotreatment of at least a portion of the deasphalted oil phase DAO obtained from step a); c) optionally, a step for catalytic cracking of at least a portion of the effluent obtained from step b).