The natural gas is deacidized by contacting in zone 3 with a solvent containing amine in aqueous solution. The gas is then dehydrated by contacting in zone 4 with a solvent very rich in amine. After the contacting operations, the solvent containing H.sub.2S. and water is expanded in drum 9, then regenerated in column 11. Part of the regenerated solvent is sent to zone 3. Another part of the regenerated solvent is vacuum distilled in column 18 so as to produce an amine-rich solvent that is sent to zone 4.