We show that the widely used Boston Mechanism (BM) fosters ability and socioeconomic segregation across otherwise identical public schools, even when schools do not have priorities over local students. Our model includes an endogenous component of school quality--determined by the peer group--and an exogenous one. If there is an exogenously worse public school, BM generates sorting of types between a priori equally good public schools: an elitist public school emerges. A richer model with some preference for closer schools and flexible residential choice does not eliminate this effect. It rather worsens the peer quality of the non-elitist school. The existence of private schools makes the best public school more elitist, while reducing the peer quality of the worst school. The main alternative assignment mechanism, Deferred Acceptance, is resilient to such sorting effects.