  We show that a Riemannian $3$-manifold with non-negative scalar curvature is
flat if it contains an area-minimizing cylinder. This scalar-curvature analogue
of the classical splitting theorem of J.~Cheeger and D.~Gromollhas been
conjectured by D.~Fischer-Colbrie and R.~Schoen and by M.~Cai and G.~Galloway.
