  We determine the Chow ring (with Q-coefficients) of M_6 by showing that all
Chow classes are tautological. In particular, all algebraic cohomology is
tautological, and the natural map from Chow to cohomology is injective. To
demonstrate the utility of these methods, we also give quick derivations of the
Chow groups of moduli spaces of curves of lower genus. The genus 6 case relies
on the particularly beautiful Brill-Noether theory in this case, and in
particular on a rank 5 vector bundle "relativizing" a baby case of a celebrated
construction of Mukai, which we interpret as a subbundle of the rank 6 vector
bundle of quadrics cutting out the canonical curve.
