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In joint work with Eva Tardos in 1995, we asked whether it was possible to obtain a polynomial-time, polylogarithmic approximation algorithm for the disjoint paths problem in the class of all even-degree planar graphs [19]. This paper answers the question in the affirmative, by providing such an algorithm. The algorithm builds on recent work of Chekuri, Khanna, and Shepherd [7, 8], who considered routing problems in planar graphs where each edge can carry up to two paths.