A cutting plane algorithm for the exact solution of the symmetric travelling salesman problem (TSP) is proposed. The real tours on a usually incomplete road network, which are in general non-Hamiltonian, are characterized directly by an integer linear programming model. The algorithm generates special cutting planes for this model. Computational results for real road networks with up to 292 visiting places are reported, as well as for classical problems of the literature with up to 120 cities. Some of the latter problems have been solved for the first time with a pure cutting plane approach.