  The latest results of the ATLAS and CMS experiments point to a preferred
narrow Higgs mass range (m_h \simeq 124 - 126 GeV) in which the effective
potential of the Standard Model (SM) develops a vacuum instability at a scale
10^{9} -10^{11} GeV, with the precise scale depending on the precise value of
the top quark mass and the strong coupling constant. Motivated by this
experimental situation, we present here a detailed investigation about the
stability of the SM^{++} vacuum, which is characterized by a simple extension
of the SM obtained by adding to the scalar sector a complex SU(2) singlet that
has the quantum numbers of the right-handed neutrino, H", and to the gauge
sector an U(1) that is broken by the vacuum expectation value of H". We derive
the complete set of renormalization group equations at one loop. We then pursue
a numerical study of the system to determine the triviality and vacuum
stability bounds, using a scan of 10^4 random set of points to fix the initial
conditions. We show that, if there is no mixing in the scalar sector, the top
Yukawa coupling drives the quartic Higgs coupling to negative values in the
ultraviolet and, as for the SM, the effective potential develops an instability
below the Planck scale. However, for a mixing angle -0.35 \alt \alpha \alt
-0.02 or 0.01 \alt \alpha \alt 0.35, with the new scalar mass in the range 500
GeV \alt m_{h"} \alt 8 TeV, the SM^{++} ground state can be absolutely stable
up to the Planck scale. These results are largely independent of TeV-scale free
parameters in the model: the mass of the non-anomalous U(1) gauge boson and its
branching fractions.
