Many networks can be usefully decomposed into a dense core plus an outlying, loosely-connected periphery. Here we propose an algorithm for performing such a decomposition on empirical network data using methods of statistical inference. Our method fits a generative model of core-periphery structure to observed data using a combination of an expectation--maximization algorithm for calculating the parameters of the model and a belief propagation algorithm for calculating the decomposition itself. We find the method to be efficient, scaling easily to networks with a million or more nodes and we test it on a range of networks, including real-world examples as well as computer-generated benchmarks, for which it successfully identifies known core-periphery structure with low error rate. We also demonstrate that the method is immune from the detectability transition observed in the related community detection problem, which prevents the detection of community structure when that structure is too weak. There is no such transition for core-periphery structure, which is detectable, albeit with some statistical error, no matter how weak it is.