The influence of oxygen on the cerium-catalyzed Belousov-Zhabotinsky reaction is investigated in a system that is closed except for transport of oxygen from the atmosphere across a free surface. Experiments show that the introduction of oxygen increases the complexity of oscillations and shortens their duration. We determine the dependence of the oscillatory pattern on the concentration of oxygen in the reaction mixture, as measured with an oxygen electrode and controlled through the content of oxygen above the free surface. We show that the influence of the stirring rate under aerobic conditions is mainly an oxygen effect and report oscillations of the oxygen concentration. Calculations based on a modified Oregonator model show that the influence of oxygen can be qualitatively accounted for through the effect of two rate constants.