(Joint work with Chris Hall.) Choose N points in CP¹, and consider ramified covers, with the simplest possible ramification above each of them, and no other ramification. Hurwitz was able to count the number of such covers. Now perturb the points; there is a corresponding perturbation of the covers, and this leads to an action of the braid group on the set of covers. The Hurwitz monodromy problem is: what finite group is the image of this action? Eisenbud, Elkies, Harris and Speiser found an upper bound on this group in the case of degree 4 covers. We will explain this and show that this upper bound is attained: it is a direct product of Sp(N-4,Z/2)'s, semidirect PSp(N-2,Z/3).