The gaseous effluent to be treated is contacted in C1 with an absorbent solution selected for its property of forming two separable phases when it has absorbed an amount of acid compounds and when it is heated.According to the invention, the absorbent solution is regenerated in at least two stages in regeneration zones Z1 and Z2. At the end of the first regeneration stage in Z1, at least part of the partly regenerated absorbent solution is separated into two fractions in B1: a fraction rich in acid compounds and a fraction depleted in acid compounds. Fraction 10 rich in acid compounds is sent to the second regeneration stage in Z2. The fraction depleted in acid compounds and regenerated absorbent solution 6 from the second regeneration stage are recycled to absorption column C1. The method can be applied to combustion fumes decarbonation and to natural gas or synthesis gas deacidizing.