  Let $\Sigma$ be a surface of constant mean curvature in ${\mathbb R}^3$ with
multiple Delaunay ends. Assuming that $\Sigma$ is non degenerate in this paper
we construct new solutions to the Cahn-Hilliard equation $\varepsilon\Delta
u+\varepsilon^{-1}u(1-u^2)=\ell_\varepsilon$ in ${\mathbb R}^3$ such that as
$\varepsilon\to 0$ the zero level set of $u_\varepsilon$ approaches $\Sigma$.
Moreover, on compacts of the connected components of ${\mathbb R}^3\setminus
\Sigma$ we have $1-|u_\varepsilon|\to 0$ uniformly.
