Acyclic edge colourings of graphs with large girth

An edge colouring of a graph $G$ is called acyclic if it is proper and every cycle contains at least three colours. We show that for every $\epsilon >0$, there exists a $g=g(\epsilon )$ such that if $G$ has girth at least $g$ then $G$ admits an acyclic edge colouring with at most $(1+\epsilon )\mathrm{\Delta }$ colours.