A process for the treatment of a gasoline containing sulphur-containing compounds, olefins and diolefins, comprising the following steps: a) fractionating the gasoline in a manner such as to recover at least one intermediate gasoline cut, MCN, comprising hydrocarbons and wherein the temperature difference (ΔT) between the 5% and 95% by weight distillation points is less than 60° C.; b) desulphurizing the intermediate gasoline cut MCN alone and in the presence of a hydrodesulphurization catalyst and hydrogen in a manner such as to produce a partially desulphurized intermediate gasoline cut MCN; and c) fractionating, in a splitter, the at least partially desulphurized intermediate gasoline cut MCN which has not undergone catalytic treatment subsequent to step b), in a manner such as to recover an intermediate gasoline with low sulphur and mercaptans contents from the column head and a cut of hydrocarbons containing sulphur-containing compounds including mercaptans from the column bottom.