A decrease in volume of the olfactory bulbs is an early marker for neurodegenerative diseases, such as Parkinson's and Alzheimer's disease. Recently, asymmetric volumes of olfactory bulbs present in postmortem MRIs of COVID-19 patients indicate that the olfactory bulbs might play an important role in the entrance of the disease in the central nervous system. Hence, volumetric assessment of the olfactory bulbs can be valuable for various conditions. Given that manual annotation of the olfactory bulbs in MRI to determine their volume is tedious, we propose a method for their automatic segmentation. To mitigate the class imbalance caused by the small volume of the olfactory bulbs, we first localize the center of each olfactory bulb in a scan using convolutional neural networks (CNNs). We use these center locations to extract a bounding box containing both olfactory bulbs. Subsequently, the slices present in the bounding box are analyzed by a segmentation CNN that classifies each voxel as left olfactory bulb, right olfactory bulb, or background. The method achieved median (IQR) Dice coefficients of 0.84 (0.08) and 0.83 (0.08), and Average Symmetrical Surface Distances of 0.12 (0.08) and 0.13 (0.08) mm for the left and the right olfactory bulb, respectively. Wilcoxon Signed Rank tests showed no significant difference between the volumes computed from the reference annotation and the automatic segmentations. Analysis took only 0.20 second per scan and the results indicate that the proposed method could be a first step towards large-scale studies analyzing pathology and morphology of the olfactory bulbs.