A process for removing hydrogen sulfide and carbon dioxide from a fluid stream comprises a) an absorption step in which the fluid stream is contacted with an absorbent comprising an aqueous solution (i) of an amine of the general formula (I) in which R1, R2 and R3 are each independently selected from C1-4-alkyl and C1-4-hydroxyalkyl; each R4 is independently selected from hydrogen, C1-4-alkyl and C1-4-hydroxyalkyl; each R5 is independently selected from hydrogen, C1-4-alkyl and C1-4-hydroxyalkyl; X is OH or NH(CR1R2R3); m is 2, 3, 4 or 5; n is 2, 3, 4 or 5; and o is 0 or 1; and optionally (ii) at least one tertiary amine, where the molar ratio of (i) to (ii) is greater than 0.05; wherein at least 90% of the hydrogen sulfide is removed from the fluid stream and selectivity for hydrogen sulfide over carbon dioxide is not greater than 8, wherein a CO2— and H2S-laden absorbent is obtained; b) a regeneration step in which at least a substream of the CO2— and H2S-laden absorbent is regenerated and a regenerated absorbent is obtained; and c) a recycling step in which at least a substream of the regenerated absorbent is recycled into the absorption step a). The process allows a high level of hydrogen sulfide removal with a simultaneously high coabsorption of carbon dioxide.