The mercaptan-laden natural gas is contacted with a mercaptan-adsorbing sieve T1. The mercaptan-rich gaseous effluent obtained upon regeneration of sieve T2 is then contacted with an olefin-containing liquid feed in the presence of an acid catalyst. Under suitable conditions, the mercaptans are absorbed in the liquid feed and they react with the olefins so as to form solvent-soluble sulfides. A solvent regeneration stage allows the capture agent to be recycled.