The fragmentation mechanism has been quite successful at providing an explanation for the formation of binary stars during the collapse phase of dense cloud cores. However, nearly all fragmentation calculations to date have ignored the effects of magnetic fields, whereas magnetic fields are generally regarded as the dominant force in molecular clouds. Here, we present the first three-dimensional, radiative hydro-dynamical models of the collapse and fragmentation of dense molecular cloud cores, including the effects of magnetic fields and ambipolar diffusion. Starting from a prolate, Gaussian cloud that would collapse and fragment in the absence of magnetic fields (a thermally supercritical cloud), we introduce sufficient magnetic field support [through the magnetic field pressure, B~2/8π, with B = B_0(ρ/ρ_0)~κ] to ensure a magnetically subcritical (stable) cloud. The effects of ambipolar diffusion are then simulated by reducing the magnetic pressure scaling factor (B_0) over a specified time interval ( = t_(AD)), which leads to a magnetically supercritical cloud and collapse. The estimated timescale for ambipolar diffusion in these dense clouds is about 10 free-fall times. The numerical models show that when t_(AD) is as long as 10 or even 20 free-fall times, fragmentation into a binary can still occur. The main effect of the magnetic field support is to delay somewhat the formation of the binary protostar. Once the dynamic collapse phase begins, a rapidly rotating, (β_i = E_(rot)/|E_(grav)| = 0.12) cloud fragments into a binary protostar. While it remains to be seen if magnetic fields can stifle fragmentation in slowly rotating clouds, rapidly rotating, magnetically supported clouds appear to be quite capable of forming binary stars.
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