A process for the elimination of acidic compounds contained in a gaseous effluent, in which a step for absorption of the acidic compounds is carried out by bringing the gaseous effluent into contact with an absorbent solution comprising: - water - at least one nitrogen-containing compound selected from di(hydroxyalkyl)monoamines having one of formulae and below: formula being in which R1 and R6 are independently selected from a hydrogen atom and an alkyl radical containing 1 to 4 carbon atoms, and R2, R3, R4 and R5 are selected without preference from a hydrogen atom and an alkyl radical containing 1 to 4 carbon atoms, with the condition that at least two of the radicals selected from R2, R3, R4 and R5 are independently selected from alkyl radicals containing 1 to 4 carbon atoms, and that two alkyl radicals are both bonded to the same carbon atom positioned alpha to the nitrogen atom, formula being in which R7 is an alkyl radical containing 1 to 4 carbon atoms, R8 and R9 are independently selected from a hydrogen atom and an alkyl radical containing 1 to 4 carbon atoms, and R10 and R11 are independently selected from a hydrogen atom and an alkyl radical containing 1 to 4 carbon atoms, with the condition that at least one of the radicals R10 or R11 is an alkyl radical containing 1 to 4 carbon atoms. - water - at least one nitrogen-containing compound selected from di(hydroxyalkyl)monoamines having one of formulae and below: formula being in which R1 and R6 are independently selected from a hydrogen atom and an alkyl radical containing 1 to 4 carbon atoms, and R2, R3, R4 and R5 are selected without preference from a hydrogen atom and an alkyl radical containing 1 to 4 carbon atoms, with the condition that at least two of the radicals selected from R2, R3, R4 and R5 are independently selected from alkyl radicals containing 1 to 4 carbon atoms, and that two alkyl radicals are both bonded to the same carbon atom positioned alpha to the nitrogen atom, formula being in which R7 is an alkyl radical containing 1 to 4 carbon atoms, R8 and R9 are independently selected from a hydrogen atom and an alkyl radical containing 1 to 4 carbon atoms, and R10 and R11 are independently selected from a hydrogen atom and an alkyl radical containing 1 to 4 carbon atoms, with the condition that at least one of the radicals R10 or R11 is an alkyl radical containing 1 to 4 carbon atoms. formula being in which R1 and R6 are independently selected from a hydrogen atom and an alkyl radical containing 1 to 4 carbon atoms, and R2, R3, R4 and R5 are selected without preference from a hydrogen atom and an alkyl radical containing 1 to 4 carbon atoms, with the condition that at least two of the radicals selected from R2, R3, R4 and R5 are independently selected from alkyl radicals containing 1 to 4 carbon atoms, and that two alkyl radicals are both bonded to the same carbon atom positioned alpha to the nitrogen atom, formula being in which R7 is an alkyl radical containing 1 to 4 carbon atoms, R8 and R9 are independently selected from a hydrogen atom and an alkyl radical containing 1 to 4 carbon atoms, and R10 and R11 are independently selected from a hydrogen atom and an alkyl radical containing 1 to 4 carbon atoms, with the condition that at least one of the radicals R10 or R11 is an alkyl radical containing 1 to 4 carbon atoms.