The periodic Toda lattice with N sites is globally symplectomorphic to a two parameter family of N-1 coupled harmonic oscillators. The action variables fill out the whole positive quadrant of ^N-1. We prove that in the interior of the positive quadrant as well as in a neighborhood of the origin, the Toda Hamiltonian is strictly convex and therefore Nekhoroshev's theorem applies on (almost) all parts of phase space.