The oscillations of the Belousov-Zhabotinskii reaction in a batch reactor vanish due to the consumption of the main reactants, i.e. bromate and malonic acid. This behavior has been analyzed first using the classical Oregonator model, where reactant concentrations are kept constant, and then introducing a modified model, in which the reactant concentration is allowed to vary according to the stoichiometry of the considered set of chemical reactions. Two different routes to the extinction of oscillations have been found, depending on whether the Oregonator exhibits a subcritical or a supercritical Hopf bifurcation. These theoretical findings are consistent with the experimental observations, that indicate that the disappearance of the oscillations takes place in two rather different ways : either suddenly after a series of practically undamped oscillations or through very slowly damped oscillations.