We present a new hypergraph stochastic blockmodel and an associated inference procedure for model-based clustering of the nodes in simple hypergraphs. Simple hypergraphs, where a node may not appear several times in a same hyperedge, have been overlooked in the literature, though they appropriately model some high-order interactions (such as co-authorship). The model assumes latent groups for the nodes and conditional independence of the hyperedges given the latent groups. We establish the first proof of generic identifiability of the parameters in such a model. We develop a variational approximation Expectation-Maximization algorithm for parameter inference and node clustering, and derive an integrated classification likelihood criterion for model selection. We illustrate the performance of our algorithm on synthetic data and analyse a real dataset of co-authorship. Our method called HyperSBM is implemented in C++ for efficiency and available as an R package at https://github.com/LB1304/HyperSBM.