Concentrated sulfuric acid and/or oleum is prepared by the catalytic oxidation of sulfur dioxide to sulfur trioxide using a gas containing water vapor, SO.sub.2 and oxygen. The catalytic oxidation is carried out in two stages. After the first stage an intermediate cooling of the reaction gas, and absorption of the water vapor content and an intermediate absorption of the sulfur trioxide formed in the first stage are performed. In the second stage the unreacted SO.sub.2 is catalytically oxidized to SO.sub.3 in the absence of water vapor, this SO.sub.3 being absorbed in sulfuric acid. An apparatus for carrying out the process is disclosed including a primary catalysis stage, means for the intermediate cooling of the reaction gas, means for the removal of the water vapor content, means for the intermediate absorption of the SO.sub.2 of the first stage, a secondary stage for the catalytic oxidation of the unreacted sulfur dioxide in the absence of water and an end absorption unit for SO.sub.2 absorption produced in the secondary stage.