A novel process is described which allows selective hydrodesulphurizing gasoline cuts containing sulphur-containing compounds and olefins. The process employs a catalyst comprising a support selected, for example, from refractory oxides such as aluminas, silicas, silica-aluminas or magnesia, used alone or as a mixture, a group VI metal, preferably Mo or W which may or may not be promoted by a group VIII metal, Co or Ni. The catalyst is characterized by a mean pore diameter of more than 22 nm. The process may comprise one or more steps.