Recent empirical evidence suggests that the Weston-Watkins support vector machine is among the best performing multiclass extensions of the binary SVM. Current state-of-the-art solvers repeatedly solve a particular subproblem approximately using an iterative strategy. In this work, we propose an algorithm that solves the subproblem exactly using a novel reparametrization of the Weston-Watkins dual problem. For linear WW-SVMs, our solver shows significant speed-up over the state-of-the-art solver when the number of classes is large. Our exact subproblem solver also allows us to prove linear convergence of the overall solver.