We prove a form of Arnold diffusion in the a priori stable case. Let H0(p) + εH1(θ, p, t), θ ∈ T n , p ∈ B n , t ∈ 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 " εH1, there exists an orbit (θ, p)(t) satisfying p(t) − p(0) > l(H1) > 0, where l(H1) is independent of ε. 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.