The natural gas arriving through pipe 1 is deacidified by being brought into contact with a solvent in zone C. The solvent charged with acid compounds is regenerated in zone R. The acid gases, released into pipe 5 upon regeneration, include a quantity of solvent. The method enables the solvent contained in the acid gases to be extracted. In zone ZA, the acid gases are brought into contact with a non-aqueous ionic liquid whose general formula is Q+ A−, where Q+ designates an ammonium, phosphonium, and/or sulfonium cation, and A− designates an anion able to form a liquid salt. The solvent is removed from the acid gases evacuated through pipe 6. The ionic liquid charged with solvent is regenerated by heating in an evaporator DE. The ionic liquid regenerated is recycled through pipes 8 and 9 to zone ZA. The solvent is evacuated through pipe 13.