Stable maps of curves and algebraic equivalence of 1-cycles

We show that algebraic equivalence of images of stable maps of curves lifts to deformation equivalence of the stable maps. The main applications concern A1(X), the group of 1-cycles modulo algebraic equivalence, for smooth, separably rationally connected varieties. If K∕k is an algebraic extension, then the kernel of A1(Xk)→A1(XK) is at most Z∕2Z. If k is finite, then the image equals the subgroup of Galois invariant cycles.