We present a single-step musical tempo estimation system based solely on a convolutional neural network (CNN). Contrary to existing systems, which typically first identify onsets or beats and then derive a tempo, our system estimates the tempo directly from a conventional melspectrogram in a single step. This is achieved by framing tempo estimation as a multi-class classification problem using a network architecture that is inspired by conventional approaches. The system's CNN has been trained with the union of three datasets covering a large variety of genres and tempi using problem-specific data augmentation techniques. Two of the three ground-truths are novel and will be released for research purposes. As input the system requires only 11.9 s of audio and is therefore suitable for local as well as global tempo estimation. When used as a global estimator, it performs as well as or better than other state-of-the-art algorithms. Especially the exact estimation of tempo without tempo octave confusion is significantly improved. As local estimator it can be used to identify and visualize tempo drift in musical performances.