Radiation treatment of cancers like prostate or cervix cancer requires considering nearby bone structures like vertebrae. In this work, we present and validate a novel automated method for the 3D segmentation of individual lumbar and thoracic vertebra in computed tomography (CT) scans. It is based on a single, low-complexity convolutional neural network (CNN) architecture which works well even if little application-specific training data are available. It is based on volume patch-based processing, enabling the handling of arbitrary scan sizes. For each patch, it performs segmentation and an estimation of up to three vertebrae center locations in one step, which enables utilizing an advanced post-processing scheme to achieve high segmentation accuracy, as required for clinical use. Overall, 1763 vertebrae were used for the performance assessment. On 26 CT scans acquired for standard radiation treatment planning, a Dice coefficient of 0.921 ± 0.047 (mean ± standard deviation) and a signed distance error of 0.271 ± 0.748 mm was achieved. On the large-sized publicly available VerSe2020 data set with 129 CT scans depicting lumbar and thoracic vertebrae, the overall Dice coefficient was 0.940 ± 0.065 and the signed distance error was 0.109 ± 0.301 mm. A comparison to other methods that have been validated on VerSe data showed that our approach achieved a better overall segmentation performance.