A monoid S is said to be right coherent if every finitely generated subact of every finitely presented right S-act is finitely presented. Left coherency is defined dually and S is coherent if it is both right and left coherent. These notions are analogous to those for a ring R (where, of course, S-acts are replaced by R-modules). Choo et al. have shown that free rings are coherent. In this paper we prove that, correspondingly, any free monoid is coherent, thus answering a question posed by Gould in 1992.