Atherosclerosis of the carotid artery is a major risk factor for stroke. Quantitative assessment of the carotid vessel wall can be based on cross-sections of three-dimensional (3D) black-blood magnetic resonance imaging (MRI). To increase reproducibility, a reliable automatic segmentation in these cross-sections is essential. We propose an automatic segmentation of the carotid artery in cross-sections perpendicular to the centerline to make the segmentation invariant to the image plane orientation and allow a correct assessment of the vessel wall thickness (VWT). We trained a residual U-Net on eight sparsely sampled cross-sections per carotid artery and evaluated if the model can segment areas that are not represented in the training data. We used 218 MRI datasets of 121 subjects that show hypertension and plaque in the ICA or CCA measuring in ultrasound. The model achieves a high mean Dice coefficient of 0.948/0.859 for the vessel's lumen/wall, a low mean Hausdorff distance of , and a low mean average contour distance of on the test set. The model reaches similar results for regions of the carotid artery that are not incorporated in the training set and on MRI of young, healthy subjects. The model also achieves a low median Hausdorff distance of on the 2021 Carotid Artery Vessel Wall Segmentation Challenge test set. The proposed method can reduce the effort for carotid artery vessel wall assessment. Together with human supervision, it can be used for clinical applications, as it allows a reliable measurement of the VWT for different patient demographics and MRI acquisition settings.