  Dynamically significant magnetic fields are routinely observed in molecular
clouds, with mass-to-flux ratio lambda = (2 pi sqrt{G}) (Sigma/B) ~ 1 (here
Sigma is the total column density and B is the field strength). It is widely
believed that ``subcritical'' clouds with lambda < 1 cannot collapse, based on
virial arguments by Mestel and Spitzer and a linear stability analysis by
Nakano and Nakamura. Here we confirm, using high resolution numerical models
that begin with a strongly supersonic velocity dispersion, that this criterion
is a fully nonlinear stability condition. All the high-resolution models with
lambda <= 0.95 form ``Spitzer sheets'' but collapse no further. All models with
lambda >= 1.02 collapse to the maximum numerically resolvable density. We also
investigate other factors determining the collapse time for supercritical
models. We show that there is a strong stochastic element in the collapse time:
models that differ only in details of their initial conditions can have
collapse times that vary by as much as a factor of 3. The collapse time cannot
be determined from just the velocity dispersion; it depends also on its
distribution. Finally, we discuss the astrophysical implications of our
results.
