Abstract: Milnor conjectured that if (M,g) is a Riemannian manifold with Rc>=0, then its fundamental group is finitely generated. Together with Brue and Semola we provide a counterexample. Specifically, we construct a manifold M^7 with Rc>=0 whose fundamental group is Q/Z . This involves a new topological construction of spaces with infinitely generated fundamental groups, and a careful analysis of the relationship between Ricci curvature and mapping class groups.