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Spinal cord tumors lead to neurological morbidity and mortality. Being able to obtain morphometric quantification (size, location, growth rate) of the tumor, edema, and cavity can result in improved monitoring and treatment planning. Such quantification requires the segmentation of these structures into three separate classes. However, manual segmentation of 3-dimensional structures is time-consuming and tedious, motivating the development of automated methods. Here, we tailor a model adapted to the spinal cord tumor segmentation task. Data were obtained from 343 patients using gadolinium-enhanced T1-weighted and T2-weighted MRI scans with cervical, thoracic, and/or lumbar coverage. The dataset includes the three most common intramedullary spinal cord tumor types: astrocytomas, ependymomas, and hemangioblastomas. The proposed approach is a cascaded architecture with U-Net-based models that segments tumors in a two-stage process: locate and label. The model first finds the spinal cord and generates bounding box coordinates. The images are cropped according to this output, leading to a reduced field of view, which mitigates class imbalance. The tumor is then segmented. The segmentation of the tumor, cavity, and edema (as a single class) reached 76.7 $\pm$ 1.5% of Dice score and the segmentation of tumors alone reached 61.8 $\pm$ 4.0% Dice score. The true positive detection rate was above 87% for tumor, edema, and cavity. To the best of our knowledge, this is the first fully automatic deep learning model for spinal cord tumor segmentation. The multiclass segmentation pipeline is available in the Spinal Cord Toolbox (https://spinalcordtoolbox.com/). It can be run with custom data on a regular computer within seconds.