We establish the existence of infinitely many complete metrics with constant scalar curvature on prescribed conformal classes on certain noncompact product manifolds. These include products of closed manifolds with constant positive scalar curvature and simply-connected symmetric spaces of noncompact or Euclidean type; in particular, $\mathbb S^m \times\mathbb R^d$, $m\geq2$, $d\geq1$, and $\mathbb S^m\times\mathbb H^d$, $2\leq d

LaTeX2e, 15 pages, revised version