A desulfurization gas process includes water vapor, CO2 and SOx (x=2 and/or 3). In a treatment unit, the gas contacts a cooled alkaline aqueous solution having a temperature lower than an initial gas temperature, water and a carbonate of an alkali metal, to cool the gas, condense some water vapor and absorb SOx in the carbonate-containing solution, produce an SOx-depleted gas and an acidic aqueous solution including sulfate and/or sulfite ions. The SOx-depleted gas and a portion of the acidic aqueous solution can then be withdrawn from the treatment unit. Carbonate of the alkali metal can be added to remaining acidic aqueous solution to obtain a made-up alkaline aqueous solution. This solution can be cooled and reused as the cooled alkaline aqueous solution. An SOx absorbent solution includes a bleed stream from a CO2-capture process, sodium or potassium carbonate, and an acidic aqueous solution obtained from desulfurization.