For applications of realistic size, both the discrete and continuous versions of the equilibrium network design problem are too computationally intensive to be solved exactly with the algorithms proposed to date. This intractibility owes to Braess' paradox which makes it necessary to constrain the flow pattern to be a noncooperative Nash or user equilibrium. This paper suggests a new heuristic for finding an approximate solution to the continuous equilibrium network design problem. Numerical tests are reported which indicate that, for networks with significant congestion, the heuristic is markedly more efficient than the Hooke-Jeeves algorithm which has been employed previously. The efficiency of the heuristic results from decomposition of the original problem into a set of interacting optimization subproblems. This decomposition is such that, at each iteration of the algorithm, only one user equilibrium needs to be calculated in order to update the improvement variables of all arcs of the network. This contrasts sharply with the Hooke-Jeeves algorithm which can require that a new user equilibrium be calculated each time an individual arc improvement variable is updated.