Adding a nonadsorbing polymer to passive colloids induces an attraction between the particles via the "depletion" mechanism. High enough polymer concentrations lead to phase separation. We combine experiments, theory, and simulations to demonstrate that using active colloids (such as motile bacteria) dramatically changes the physics of such mixtures. First, significantly stronger interparticle attraction is needed to cause phase separation. Secondly, the finite size aggregates formed at lower interparticle attraction show unidirectional rotation. These micro-rotors demonstrate the self-assembly of functional structures using active particles. The angular speed of the rotating clusters scales approximately as the inverse of their size, which may be understood theoretically by assuming that the torques exerted by the outermost bacteria in a cluster add up randomly. Our simulations suggest that both the suppression of phase separation and the self-assembly of rotors are generic features of aggregating swimmers and should therefore occur in a variety of biological and synthetic active particle systems.