Over the last decades, several studies have demonstrated the importance of co-clustering to simultaneously produce groups of objects and features. Even to obtain object clusters only, using co-clustering is often more effective than one-way clustering, especially when considering sparse high dimensional data. In this paper, we present a novel generative mixture model for co-clustering such data. This model, the Sparse Poisson Latent Block Model (SPLBM), is based on the Poisson distribution, which arises naturally for contingency tables, such as document-term matrices. The advantages of SPLBM are two-fold. First, it is a rigorous statistical model which is also very parsimonious. Second, it has been designed from the ground up to deal with data sparsity problems. As a consequence, in addition to seeking homogeneous blocks, as other available algorithms, it also filters out homogeneous but noisy ones due to the sparsity of the data. Experiments on various datasets of different size and structure show that an algorithm based on SPLBM clearly outperforms state-of-the-art algorithms. Most notably, the SPLBM-based algorithm presented here succeeds in retrieving the natural cluster structure of difficult, unbalanced datasets which other known algorithms are unable to handle effectively.