NOVELTY - Deacidification of a gaseous effluent comprising acid compound(s) of hydrogen sulfide, mercaptans, carbon dioxide, carbon oxysulfide, sulfur dioxide, and/or carbon disulfide, using an absorbent solution comprising reactive compounds with greater than or equal to 2 amine functions, comprises regenerating a second liquid phase enriched in acid compounds and laden with more acid molecules, to release a gas stream rich in acid compounds and a liquid stream poor in acid compounds and laden with more acid molecules.    USE - For deacidification of a gaseous effluent (comprising acid compound(s) of H2S, mercaptans, CO2, COS, SO2, and/or CS2), e.g. combustion fumes, natural gas or conversion gas, Claus tail gas, gas resulting from biomass fermentation, cement works gas or blast-furnace gas, or refinery gas.    ADVANTAGE - The method reduces the amount of energy required for regenerating an absorbent solution laden with acid compounds. It regenerates only 43 wt.% of the absorbent solution used.    DETAILED DESCRIPTION - Deacidification of a gaseous effluent comprising acid compound(s) of hydrogen sulfide (H2S), mercaptans, carbon dioxide (CO2), COS, sulfur dioxide (SO2), and/or CS2, using an absorbent solution comprising reactive compounds with greater than or equal to 2 amine functions, comprises contacting the gaseous effluent with a first fraction of the absorbent solution, to obtain a gaseous effluent depleted in acid compounds and a liquid effluent laden with acid compounds; contacting the liquid effluent obtained, with a second fraction of absorbent solution laden with molecules more acid than the acid compounds, to cause formation of a first liquid phase depleted in acid compounds and of a second liquid phase enriched in acid compounds and laden with more acid molecules; separating the first phase from the second phase; and regenerating the second phase to release a gas stream rich in acid compounds and a liquid stream poor in acid compounds and laden with more acid molecules.