This paper presents a hybrid algorithm that combines local search and constraint programming techniques to solve a network routing problem. The problem considered is that of routing traffic demands from a set of requests over a network with limited capacity so as to minimise the cost of any unrouted demands. The hybridisation is twofold: pure local search is used to find a good cost bound for a subsequent branch-and-bound optimisation phase, with local search again applied at the nodes of the branch-andbound search tree. Constraint propagation occurs in the search tree to reduce the domains of the decision variables, using a set of constraints that are independent of the action of local search at the nodes. In contrast to previous constraint programming/local search hybridisations, here local search is used to satisfy the hard problem constraints, while optimisation is handled in the framework of constraint programming. The resulting algorithm is incomplete, but is shown to be compare favourably with a complete approach to this problem.