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It is shown that the four-gluon scattering amplitude for N = 4 supersymmetric Yang-Mills theory in the planar limit can be written, in both the weak- and strong-coupling limits, as a reggeized amplitude, with a parent trajectory and an infinite number of daughter trajectories. This result is not evident a priori, and relies crucially on the fact that the leading IR-divergence and the finite log2 (s / t)-dependent piece of the amplitude are characterized by the same function for all values of the coupling, as conjectured by Bern, Dixon, and Smirnov, and proved by Alday and Maldacena in the strong-coupling limit. We use the Alday-Maldacena result to determine the exact strong-coupling Regge trajectory. © 2007 Elsevier B.V. All rights reserved.