Information about clouds is important for observing and predicting weather and climate as well as for generating and distributing solar power. Most existing approaches extract cloud information from satellite data by classifying individual pixels instead of using closely integrated spatial information, ignoring the fact that clouds are highly dynamic, spatially continuous entities. This paper proposes a novel cloud classification method based on deep learning. Relying on a Convolutional Neural Network (CNN) architecture for image segmentation, the presented Cloud Segmentation CNN (CS-CNN), classifies all pixels of a scene simultaneously rather than individually. We show that CS-CNN can successfully process multispectral satellite data to classify continuous phenomena such as highly dynamic clouds. The proposed approach produces excellent results on Meteosat Second Generation (MSG) satellite data in terms of quality, robustness, and runtime compared to other machine learning methods such as random forests. In particular, comparing CS-CNN with the CLAAS-2 cloud mask derived from MSG data shows high accuracy (0.94) and Heidke Skill Score (0.90) values. In contrast to a random forest, CS-CNN produces robust results and is insensitive to challenges created by coast lines and bright (sand) surface areas. Using GPU acceleration, CS-CNN requires only 25 ms of computation time for classification of images of Europe with $508\times 508$ pixels.