The natural gas arriving through pipe 1 is deacidified by being brought into contact with a solvent in column C2. The solvent charged with acid compounds is regenerated in zone R. The purified gas evacuated by pipe 9 includes some of the solvent. The method enables the solvent contained in the purified gas to be extracted. In zone ZA, the purified gas is brought into contact with a non-aqueous ionic liquid whose general formula is Q.sup.+ A.sup.-, where Q.sup.+ designates an ammonium, phosphonium, and/or sulfonium cation, and A.sup.- designates an anion able to form a liquid salt. The solvent-impoverished purified gas is evacuated through pipe 17. The ionic liquid charged with solvent is regenerated by heating in an evaporator DE. The solvents separated from the ionic liquid in evaporator DE are recycled.