The combustion fume flowing in through line 1 is decarbonated by contacting with a solvent in column C2. The solvent laden with carbon dioxide is regenerated in zone R. The purified fume discharged through line 9 comprises part of the solvent. The method allows to extract the solvent contained in the purified fume. The purified fume is contacted in zone ZA with a non-aqueous ionic liquid of general formula Q.sup.+ A.sup.-; Q.sup.+ designates an ammonium, phosphonium and/or sulfonium cation, and A.sup.- an anion likely to form a liquid salt. The solvent-depleted purified fume is discharged through line 17. The solvent-laden ionic liquid is regenerated by heating in evaporation device DE. The solvent separated from the ionic liquid in device DE is recycled.