This paper proposes a new hybrid algorithm for solving the Arc Routing Problem with Stochastic Demands (ARPSD). Our approach combines Monte Carlo simulation (MCS) with the RandSHARP algorithm, which is designed for solving the Capacitated Arc Routing Problem (CARP) with deterministic demands. The RandSHARP algorithm makes use of a CARP-adapted version of the Clarke and Wright Savings heuristic, which was originally designed for the Vehicle Routing Problem. The RandSHARP algorithm also integrates a biased-randomized process, which allows it to obtain competitive results for the CARP in low computational times. The RandSHARP algorithm is then combined with MCS to solve the ARPSD. Some numerical experiments contribute to illustrate the potential benefits of our approach.