Applying machine learning models to meteorological data brings many opportunities to the Geosciences field, such as predicting future weather conditions more accurately. In recent years, modeling meteorological data with deep neural networks has become a relevant area of investigation. These works apply either recurrent neural networks (RNN) or some hybrid approach mixing RNN and convolutional neural networks (CNN). In this work, we propose STConvS2S (Spatiotemporal Convolutional Sequence to Sequence Network), a deep learning architecture built for learning both spatial and temporal data dependencies using only convolutional layers. Our proposed architecture resolves two limitations of convolutional networks to predict sequences using historical data: (1) they violate the temporal order during the learning process and (2) they require the lengths of the input and output sequences to be equal. Computational experiments using air temperature and rainfall data from South America show that our architecture captures spatiotemporal context and that it outperforms or matches the results of state-of-the-art architectures for forecasting tasks. In particular, one of the variants of our proposed architecture is 23% better at predicting future sequences and five times faster at training than the RNN-based model used as a baseline.