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A method is developed for constructing the conformal map of a distorted region onto a rectangle. A discrete Fourier transform is used to map the boundary of the region onto the boundary of the rectangle; the resulting equations may be solved using a fast Fourier transform algorithm. The map for internal points may then be constructed using a standard Laplace equation solver. The method is computationally competitive, and is applicable to field problems, for instance in fluid mechanics. 1