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We model the self-initiated formation and contraction of protostellar cores due to ambipolar diffusion in axisymmetric, self-gravitating, isothermal, magnetic molecular clouds, accounting for a cosmic abundance of interstellar grains (both charged and neutral) and an external ultraviolet radiation field. The evolution of model clouds is followed numerically to a central density enhancement of $10^6$ (e.g., from 2.6 \times 10^3$ to $2.6 \times 10^9~\cc$). First, ambipolar diffusion slowly increases the mass-to-flux ratio of a cloud's central flux tubes, leading to the formation of thermally supercritical but magnetically subcritical cores. The time scale for this process is essentially the initial central flux-loss time scale, which exceeds the dynamical time scale ($\simeq$ free-fall time scale) typically by a factor $10-20$. Eventually, the mass-to-flux ratio exceeds the critical value for collapse. The subsequent contraction of the thermally and magnetically supercritical cores becomes progressively more dynamic, while the envelopes remain relatively well supported by magnetic forces, in agreement with early theoretical predictions by Mouschovias. A typical supercritical core consists of a uniform-density central region and a ``tail" of infalling matter with a power-law density profile $\nn \propto r^s$, $-1.50 \simgt s \simgt -1.85$. The mass infall (or accretion) rate is controlled by ambipolar diffusion, and differs qualitatively and quantitatively from estimates based on nonmagnetic calculations and their extrapolations to magnetic clouds. Model clouds that include the macroscopic (collisional) effects of grains have their evolution retarded (typically by $50\%$) with respect to models accounting only for neutral-ion drag. Neutral-grain drag typically dominates the neutral-ion drag at densities $\nn \simgt 10^8~\cc$. UV radiation increases the degree of ionization in model cloud envelopes typically by a factor of 100 compared to models accounting only for cosmic-ray ionization. As a result, UV ionization significantly decreases the ambipolar-diffusion controlled mass infall rate in molecular cloud envelopes.
