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We formulate and solve the problem of ambipolar-diffusion and/or magnetic-braking initiated formation and contraction of protostellar cores in self-gravitating, magnetically supported, rotating, isothermal model molecular clouds. If it were not for ambipolar diffusion and magnetic braking, the model clouds, which are oblate, aligned rotators (J parallel B) initially in exact equilibrium states (with gravitational and external-pressure forces balanced by magnetic, thermal-pressure, and centrifugal forces), would undergo no evolution at all. We follow the evolution up to six orders of magnitude enhancement of the central density (e.g., from 3 x 10<SUP>3</SUP> to 3 x 10<SUP>9</SUP>/ cu cm). A thermally and magnetically supercritical core forms because of ambipolar diffusion. It is characterized by a compact, uniform-density central region, shrinking both in size and mass, surrounded by a 'tail' of matter left behind, in which a near power-law density profile is established, as found earlier by Mouschovias et al. in the absence of magnetic braking. The envelope remains well supported by magnetic forces. Magnetic braking leads to a very rapid decrease of the specific angular momentum everywhere in a model cloud, so that centrifugal forces play a negligible role in the formation and evolution of protostellar cores, at least up to central densities of approximately 3 x 10<SUP>9</SUP>/cu cm. Typically, the evolution of the angular velocity Omega<SUB>c</SUB> of the compact, uniform-density central part of the core exhibits three distinct phases: (1) an exponential decrease of Omega<SUB>c</SUB>(t) due to effective magnetic braking, before ambipolar diffusion induces any significant redistribution of mass in the central flux tubes; (2) a constant-Omega<SUB>c</SUB> phase, which lasts until a magnetically supercritical core forms in the subcritical cloud; and (3) a constant angular momentum (J) phase, coinciding with the rapid contraction phase of the supercritical core. We predict the typical structure, mass, size, magnetic field strength, angular momentum, etc., of protostellar cores; for example, by the end of a typical run, a core's specific angular momentum (J/M) is reduced by two orders of magnitude, to a value comparable to that of wide binary systems. Moreover we find that, even in the absence of magnetic braking, the gravitational field of the nonhomologously contracting, oblate central region increases just as rapidly as the centrifugal acceleration proportional to r<SUP>-3</SUP>, so that centrifugal forces do not become important during these stages of evolution.