  We prove a form of Arnold diffusion in the a priori stable case. Let H0(p) +
$\epsilon$H1($\theta$, p, t), $\theta$ $\in$ T n , p $\in$ B n , t $\in$ T =
R/T be a nearly integrable system of arbitrary degrees of freedom n 2 with a
strictly convex H0. We show that for a "generic" $\epsilon$H1, there exists an
orbit ($\theta$, p)(t) satisfying p(t) -- p(0) {\textgreater} l(H1)
{\textgreater} 0, where l(H1) is independent of $\epsilon$. The diffusion orbit
travels along a co-dimension one resonance , and the only obstruction to our
construction is a finite set of additional resonances. For the proof we use a
combination geometric and variational methods, and manage to adapt tools which
have recently been developed in the a priori unstable case.
