  We prove that every closed, universally embeddable CR three-manifold with
nonnegative Yamabe constant and positive total $Q^\prime$-curvature is contact
diffeomorphic to a quotient of the standard contact three-sphere. We also prove
that every closed, embeddable CR three-manifold with zero Yamabe constant and
nonnegative total $Q^\prime$-curvature is CR equivalent to a compact quotient
of the Heisenberg group with its flat CR structure.
