A cutting-plane type algorithm for solving entropy optimization problems with a finite number of variables but an infinite number of linear constraints is proposed in this paper. In each iteration, we solve a finite entropy optimization problem and add one more constraint. The iterative process ends when an optimal solution is identified. A convergence proof, under some mild conditions, is given. An efficient implementation based on a dual approach is also included. Our preliminary computational experience confirms the efficiency of the proposed method and shows its potential for solving large-scale problems.