  An \emph{acyclic edge-coloring} of a graph $G$ is a proper edge-coloring of
$G$ such that the subgraph induced by any two color classes is acyclic. The
\emph{acyclic chromatic index}, $\chi'_a(G)$, is the smallest number of colors
allowing an acyclic edge-coloring of $G$. Clearly $\chi'_a(G)\ge \Delta(G)$ for
every graph $G$. Cohen, Havet, and M\"{u}ller conjectured that there exists a
constant $M$ such that every planar graph with $\Delta(G)\ge M$ has
$\chi'_a(G)=\Delta(G)$. We prove this conjecture.
