  We address the issue of the renormalizability of the gauge-invariant
non-local dimension-two operator $A^2_{\rm min}$, whose minimization is defined
along the gauge orbit. Despite its non-local character, we show that the
operator $A^2_{\rm min}$ can be cast in local form through the introduction of
an auxiliary Stueckelberg field. The localization procedure gives rise to an
unconventional kind of Stueckelberg-type action which turns out to be
renormalizable to all orders of perturbation theory. In particular, as a
consequence of its gauge invariance, the anomalous dimension of the operator
$A^2_{\rm min}$ turns out to be independent from the gauge parameter $\alpha$
entering the gauge-fixing condition, being thus given by the anomalous
dimension of the operator $A^2$ in the Landau gauge.
