We present a new model of molecular cloud cores that originate from
filamentary clouds that are threaded by helical magnetic fields. Only modest
toroidal fields are required to produce elongated cores, with intrinsic axis
ratios in the range 0.1 <= q <= 0.9 and mean projected axis ratios in the range
<= 1. Thus many of our models are in good agreement with the observed shapes of cores (Myers et al 1991, Ryden 1996), which find axis ratios distributed about the mean values
\approx 0.5-0.6 and q \approx 0.4-0.5. We find that the Bonnor-Ebert critical mass is reduced by about 20% by the helical field threading our models. We also find that models are generally most elongated when the mass is significantly less than the Bonnor-Ebert critical mass for unmagnetized cores.