We consider Gevrey perturbations H of a completely integrable non-degenerate Gevrey Hamiltonian H0. Given a Cantor set $\Omega_\kappa$ defined by a Diophantine condition, we find a family of Kolmogorov–Arnold–Moser (KAM) invariant tori of H with frequencies $\omega\in \Omega_\kappa$ which is Gevrey smooth in a Whitney sense. Moreover, we obtain a symplectic Gevrey normal form of the Hamiltonian in a neighborhood of the union $\Lambda$ of the invariant tori. This leads to effective stability of the quasi-periodic motion near $\Lambda$.