  We revisit the problem of the formation of dense protostellar cores due to
ambipolar diffusion within magnetically supported molecular clouds, and derive
an analytical expression for the core formation timescale. The resulting
expression is similar to the canonical expression = t_{ff}^2/t_{ni} ~ 10 t_{ni}
(where t_{ff} is the free-fall time and t_{ni} is the neutral-ion collision
time), except that it is multiplied by a numerical factor C(\mu_{c0}), where
\mu_{c0} is the initial central mass-to-flux ratio normalized to the critical
value for gravitational collapse. C(\mu_{c0}) is typically ~ 1 in highly
subcritical clouds (\mu_{c0} << 1), although certain conditions allow
C(\mu_{c0}) >> 1. For clouds that are not highly subcritical, C(\mu_{c0}) can
be much less than unity, with C(\mu_{c0}) --> 0 for \mu_{c0} --> 1,
significantly reducing the time required to form a supercritical core. This,
along with recent observations of clouds with mass-to-flux ratios close to the
critical value, may reconcile the results of ambipolar diffusion models with
statistical analyses of cores and YSO's which suggest an evolutionary timescale
\~ 1 Myr for objects of mean density ~ 10^4 cm^{-3}. We compare our analytical
relation to the results of numerical simulations, and also discuss the effects
of dust grains on the core formation timescale.
