We introduce the task of action-driven stochastic human motion prediction, which aims to predict multiple plausible future motions given a sequence of action labels and a short motion history. This differs from existing works, which predict motions that either do not respect any specific action category, or follow a single action label. In particular, addressing this task requires tackling two challenges: The transitions between the different actions must be smooth; the length of the predicted motion depends on the action sequence and varies significantly across samples. As we cannot realistically expect training data to cover sufficiently diverse action transitions and motion lengths, we propose an effective training strategy consisting of combining multiple motions from different actions and introducing a weak form of supervision to encourage smooth transitions. We then design a VAE-based model conditioned on both the observed motion and the action label sequence, allowing us to generate multiple plausible future motions of varying length. We illustrate the generality of our approach by exploring its use with two different temporal encoding models, namely RNNs and Transformers. Our approach outperforms baseline models constructed by adapting state-of-the-art single action-conditioned motion generation methods and stochastic human motion prediction approaches to our new task of action-driven stochastic motion prediction. Our code is available at https://github.com/weimao-2019/WAT.