The traffic assignment problem with elastic demands can be formulated as an optimization problem, whose objective is sum of a congestion function and a disutility function. We propose to use a variant of the Analytic Center Cutting Plane Method to solve this problem. We test the method on instances with different congestion functions (linear with capacities on the arc and BPR) and different demand functions (constant elasticity and linear). The results of the numerical experiments show that it is possible to solve large instances with high accuracy.