  In Paper I (Fiege & Pudritz, 1999), we constructed models of filamentary
molecular clouds that are truncated by a realistic external pressure and
contain a rather general helical magnetic field. We address the stability of
our models to gravitational fragmentation and axisymmetric MHD-driven
instabilities. By calculating the dominant modes of axisymmetric instability,
we determine the dominant length scales and growth rates for fragmentation. We
find that the role of pressure truncation is to decrease the growth rate of
gravitational instabilities by decreasing the self-gravitating mass per unit
length. Purely poloidal and toroidal fields also help to stabilize filamentary
clouds against fragmentation. The overall effect of helical fields is to
stabilize gravity-driven modes, so that the growth rates are significantly
reduced below what is expected for unmagnetized clouds. However, MHD
``sausage'' instabilities are triggered in models whose toroidal flux to mass
ratio exceeds the poloidal flux to mass ratio by more than a factor of $\sim
2$. We find that observed filaments appear to lie in a physical regime where
the growth rates of both gravitational fragmentation and axisymmetric
MHD-driven modes are at a minimum.
