For desulfurization of a hydrocarbon-containing feedstock, at least one stage for selective hydrogenation of diolefins present in the initial hydrocarbon feedstock, in the presence of a catalyst of group VIII of the period table, and a stage for extraction by a solvent of the resultant hydrogenated fraction under conditions that provide at least two fractions: a raffinate that comprises for the most part olefins, paraffins and naphthenes and a reduced amount of sulfur-containing compounds that are contained in the initial feedstock, a fraction that contains the majority of the aromatic hydrocarbons and the majority of the sulfur-containing compounds contained in the initial feedstock.