  We present accurate predictions of the effective Majorana mass
$|m_{\beta\beta}|$ in neutrinoless double-$\beta$ decay in the standard case of
$3\nu$ mixing and in the case of 3+1 neutrino mixing indicated by the reactor,
Gallium and LSND anomalies. We have taken into account the uncertainties of the
neutrino mixing parameters determined by oscillation experiments. It is shown
that the predictions for $|m_{\beta\beta}|$ in the cases of $3\nu$ and 3+1
mixing are quite different, in agreement with previous discussions in the
literature, and that future measurements of neutrinoless double-$\beta$ decay
and of the effective light neutrino mass in $\beta$ decay or the total mass of
the three lightest neutrinos in cosmological experiments may distinguish the
$3\nu$ and 3+1 cases if the mass ordering is determined by oscillation
experiments. We also present a relatively simple method to determine the
minimum value of $|m_{\beta\beta}|$ in the general case of $N$-neutrino mixing.
