An acid gas such as carbon dioxide, hydrogen sulfide, or a mixture thereof is removed from gaseous streams using aqueous absorption and stripping processes. By replacing the conventional stripper used to regenerate the aqueous solvent and capture the acid gas with a multipressure stripper (51) that combines acid gas compression with stripping, less energy is consumed. The multipressure stripper is a multistage flash (52, 55, 59) in which the total vapor flow from each stage is compressed and fed to the bottom of the previous flash stage at a higher pressure. In this process, the heat in the water content of the vapor exiting each stage is utilized at a higher pressure in the previous stage. The described stripping process generates the acid gas at a higher pressure without operating the stripper at a higher temperature, thereby reducing the energy consumption of the system.