An acyclic edge colouring of a graph is a proper edge colouring having no 2-coloured cycle, that is, a colouring in which the union of any two colour classes forms a linear forest. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge colouring using k colours and is usually denoted by a′(G). Determining a′(G) exactly is a very hard problem (both theoretically and algorithmically) and is not determined even for complete graphs. We show that a′(G) ≤ Δ(G) + 1, if G is an outerplanar graph. This bound is tight within an additive factor of 1 from optimality. Our proof is constructive leading to an O(n log Δ) time algorithm. Here, Δ = Δ(G) denotes the maximum degree of the input graph. © Springer-Verlag Berlin Heidelberg 2007.