Let X be a finite set and let G be a finite group acting on X. The group action splits X into disjoint orbits. The Burnside process is a Markov chain on X which has a uniform stationary distribution when the chain is lumped to orbits. We consider the case where X = [k] n with k ≥ n and G = S k is the symmetric group on [k], such that G acts on X by permuting the value of each coordinate. The resulting Burnside process gives a novel algorithm for sampling a set partition of [n] uniformly at random. We obtain bounds on the mixing time and show that the chain is rapidly mixing in the number of orbits.