Use is made, for selectively hydrogenating the olefinic double bonds of block copolymers, at least one block of which comprises olefinic double bonds, using a catalyst based on a metal from Group VIII in a medium comprising an organic solvent for the copolymer and an ionic liquid as solvent for the catalyst, of a water-immiscible ionic liquid, preferably an ionic liquid for which the anion is the hexafluorophosphate anion and the cation is the 1-butyl-3-methylimidazolium (bmim.sup.+) or 1-ethyl-3-methylimidazolium (emim.sup.+) cation.Applied to poly(styrene)-b-poly(butadiene)-b-poly(methyl methacrylate) block copolymers, the poly(butadiene) block of which predominantly possesses a 1,4-microstructure, this process results in copolymers for which the degree of hydrogenation is at least equal to 50% and which exhibit a melting point of greater than 30 .degree.C.