In this paper we deal with Prandtl's boundary layer problem for incompressible laminar flow past a wedge. When the Reynolds number is large the solution of this problem has a parabolic boundary layer. We construct a direct numerical method for computing approximations to the solution of this problem using a piecewise uniform fitted mesh technique appropriate to the parabolic boundary layer. We use the numerical method to approximate the self--similar solution of Prandtl's problem in a finite rectangle excluding the leading edge of the wedge, which is the source of an additional singularity caused by incompatibility of the problem data. We verify that the constructed numerical method is robust in the sense that the computed errors for the velocity components and their derivatives in the discrete maximum norm are Reynolds uniform. We construct and apply a special numerical method related to the Falkner--Skan technique to compute a reference solution for the error analysis of the velocity components and their derivatives. By means of extensive numerical experiments we show that the constructed direct numerical method is Reynolds--uniform.