The raw natural gas is deacidized and dehydrated in units DA and DH. The treated gas is then purified by adsorption of the mercaptans in first enclosure A1. Part of the purified gas is heated in E1, then fed into second enclosure A2 so as to discharge the water adsorbed by the adsorbent material contained in this second enclosure. A steam-rich stream is fed into third enclosure A3 containing a mercaptan-laden adsorbent material. In A3, the mercaptans are desorbed and replaced by the steam.