A numerical procedure to determine the discrete conformal mapping of an arbitrary simply connected region onto the open unit disk is described. The method is fast and directly provides an estimate of the global error due to the discretization of the mapping. The intimate relationship between the Riemann Mapping Theorem and the numerical method of construction of the conformal mapping is brought out in the presentation. The procedure is an interesting example of a method of construction that is directly based on a theorem guaranteeing existence and uniqueness.