  We introduce the moduli space of spectral curves of constant mean curvature
(\cmc\hspace{-5pt}) cylinders of finite type in the round unit 3-sphere. The
subset of spectral curves of mean-convex Alexandrov embedded cylinders is
explicitly determined using a combination of integrable systems and geometric
analysis techniques. We prove that these cylinders are surfaces of revolution.
As a consequence all mean-convex Alexandrov embedded {\sc{cmc}} tori in the
3-sphere are surfaces of revolution.
