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}, $χ'_a(G)$, is the smallest number of colors allowing an acyclic edge-coloring of $G$. Clearly $χ'_a(G)\ge Δ(G)$ for every graph $G$. Cohen, Havet, and Müller conjectured that there exists a constant $M$ such that every planar graph with $Δ(G)\ge M$ has $χ'_a(G)=Δ(G)$. We prove this conjecture.