  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 $\varepsilon>0$,
there exists a $g=g(\varepsilon)$ such that if $G$ has girth at least $g$ then
$G$ admits an acyclic edge colouring with at most $(1+\varepsilon)\Delta$
colours.
