  We present a renormalization group study of the scalar potential in a minimal
$U(1)_{B-L}$ extension of the Standard Model involving one extra heavier Higgs
and three heavy right-handed neutrinos with family universal B-L charge
assignments. We implement a type-I seesaw for the masses of the light neutrinos
of the Standard Model. In particular, compared to a previous study, we perform
a two-loop extension of the evolution, showing that two-loop effects are
essential for the study of the stability of the scalar potential up to the
Planck scale. The analysis includes the contribution of the kinetic mixing
between the two abelian gauge groups, which is radiatively generated by the
evolution, and the one-loop matching conditions at the electroweak scale. By
requiring the stability of the potential up to the Planck mass, significant
constraints on the masses of the heavy neutrinos, on the gauge couplings and
the mixing in the Higgs sector are identified.
