The invention relates to a process for removing hydrogen sulfide and carbon dioxide from a feed gas stream. H2S in the feed gas stream is converted to elemental sulfur in a Claus unit. At least a partof the gas stream obtained is contacted with an aqueous lean absorbing medium in an absorption zone at a pressure between 0.9 and 2 bara. The aqueous lean absorbing medium used comprises one or moreamines chosen from: a polyamine in the absence of tertiary amine functionalities having a pKa sufficient to neutralize carbamic acid, the polyamine having at least one primary amine functionality having a pKa smaller than 10.0 at 25 DEG C, a polyamine in the absence of tertiary amine functionalities having a pKa sufficient to neutralize carbamic acid, the polyamine having at least one secondary amine functionality having a pKa for each sorbing nitrogen smaller than 10.0 at 25 DEG C. The process is improved as compared to a process involving Claus off-gas treatment with (activated) MDEA. Effective CO2 removal is achieved while at the same time a simplified line-up with less equipment can be used.