The algorithm described in this paper discovers one or more motifs in a collection of DNA or protein sequences by using the technique of expectation maximization to fit a two-component finite mixture model to the set of sequences. Multiple motifs are found by fitting a two-component finite mixture model to the data, probabilistically erasing the occurrences of the motif thus found, and repeating the process to find successive motifs. The algorithm requires only a set of sequences and a number specifying the width of the motifs as input. It returns a model of each motif and a threshold which together can be used as a Bayes-optimal classifier for searching for occurrences of the motif in other databases. The algorithm estimates how many times each motif occurs in the input dataset and outputs an alignment of the occurrences of the motif. The algorithm is capable of discovering several different motifs with differing numbers of occurrences in a single dataset. Motifs are discovered by treating the set of sequences as though they were created by a stochastic process which can be modelled as a mixture of two densities, one of which generated the occurrences of the motif, and the other the rest of the positions in the sequences. Expectation maximization is used to estimate the parameters of the two densities and the mixing parameter.