NOVELTY - Process for eliminating acid compounds contained in a gaseous effluent, comprises absorption step of acid compounds by contacting the effluent with an absorbent solution comprising water, and at least one triamine compound (I) comprising two tertiary amine function and secondary amine function.    USE - The process is useful for eliminating acid compounds contained in a gaseous effluent, where the gaseous effluent is natural gas, synthesis gas, flue gas, refinery gas, gas obtained in Claus tail process, gas from biomass fermentation, cement gas, and smoke from incinerator. The process is useful for the selective elimination of hydrogen sulfide from a gaseous effluent comprising hydrogen sulfide and carbon dioxide (all claimed).    ADVANTAGE - The solvent used in the process has high cyclic capacity and low enthalpy of binding. The deacidification process can be carried out at lower operating costs (including energy costs for regeneration and circulation of the solvent).    DETAILED DESCRIPTION - Process for eliminating acid compounds contained in a gaseous effluent, comprises absorption step of acid compounds by contacting the effluent with an absorbent solution comprising water, and at least one triamine compound of formula (X-(CR7R8)a-CR1R2-NH-CR4R3-(-CR6R5-)b-Y1) (I) comprising two tertiary amine function and secondary amine function.    R1-R8 = H, 1-6C alkyl or 1-6C alkylene;    a, b = 1-5;    X, Y1 = amine group of formula (-NR9R10) (A) or amine group of formula (B);    R9, R10 = 1-6C alkyl hydrocarbon or 1-6C alkylene;    R11, R12 = 1-6C alkyl hydrocarbon, 1-6C alkylene or H;    Z = an ether or a covalent bond; and    x, y = 1-3.    Provided that: X and Y1 are each in the definition of B1; X responds to the definition of A and Y1 responds to the definition of B1; X responds to the definition of B1 and Y1 responds the definition of A; or X and Y1 are each in the definition of A and at least one of the four groups of R1, R2, R3 and R4 is 1-6C alkyl hydrocarbon or 1-6C alkylene. An INDEPENDENT CLAIM is included for a process for treating a gas comprising selective elimination of hydrogen sulfide (H2S) from a gas effluent comprising (H2S) and carbon dioxide (CO2).