In this paper, we propose an exact algorithm to solve the Orthogonal Art Gallery problem in which guards can only be placed on the vertices of the polygon P representing the gallery. Our approach is based on a discretization of P into a finite set of points in its interior. The algorithm repeatedly solves an instance of the Set Cover problem obtaining a minimum set Z of vertices of P that can view all points in the current discretization. Whenever P is completely visible from Z, the algorithm halts; otherwise, the discretization is refined and another iteration takes place. We establish that the algorithm always converges to an optimal solution by presenting a worst case analysis of the number of iterations that could be effected. Even though these could theoretically reach O(n 4 ), our computational experiments reveal that, in practice, they are linear in n and, for n ≤ 200, they actually remain less than three in almost all instances.Furthermore, the low number of points in the initial discretization, O(n 2 ), compared to the possible O(n 4 ) atomic visibility polygons, renders much shorter total execution times. Optimal solutions found for different classes of instances of polygons with up to 200 vertices are also described.