We describe the Chow ring with rational coefficients of the moduli space of stable maps with marked points M0,m(P , d) as the subring of invariants of a ring B(M0,m(P , d);Q), relative to the action of the group of symmetries Sd. B (M0,m(P , d);Q) is computed by following a sequence of intermediate spaces for M0,m(P , d) and relating them to substrata of M0,1(P , d + m − 1). An additive basis f...