We study the Lipschitz metric on Outer Space and prove that fully irreducible
elements of Out(F_n) act by hyperbolic isometries with axes which are strongly
contracting. As a corollary, we prove that the axes of fully irreducible
automorphisms in the Cayley graph of Out(F_n) are stable, meaning that a
quasi-geodesic with endpoints on the axis stays within a bounded distance from
the axis.