A process for desulphurizing a gasoline cut containing olefins, sulphur-containing compounds and optionally molecules belonging to C3 and C4 cuts comprises at least a first step A for contacting said gasoline cut with an acidic resin having an acid capacity of more than 4.7 equivalents per kg and a specific surface area of less than 55 m.sup.2/g, and a second step B for fractionation of the mixture from the first step.