  A modification of the Abelian Duality transformations is proposed
guaranteeing that a (not necessarily conformally invariant) $\sigma$-model be
quantum equivalent (at least up to two loops in perturbation theory) to its
dual. This requires a somewhat non standard perturbative treatment of the {\sl
dual} $\sigma$-model. Explicit formulae of the modified duality transformation
are presented for a special class of block diagonal purely metric
$\sigma$-models.
