The problem of the formation and contraction of fragments (or cores) in magnetically supported parent molecular clouds was formulated in a previous paper. Three dimensionless free parameters appear in the evolution equations: the initial ratio of the free-fall and neutral-ion collision times (in the uniform reference state), nu(ff, 0), the exponent k in the relation between the ion and neutral densities n(i) is-proportional-to n(n)k, and the initial ratio of the magnetic and thermal pressures, alpha0. The initial central mass-to-flux ratio in units of the critical value for gravitational collapse, mu0, enters through the initial conditions. We follow both the quasistatic and dynamic phases of contraction and demonstrate that ambipolar diffusion leads to self-initiated protostar formation (''quasistatic'' meaning motion with negligible acceleration). A typical cloud core forms and contracts quasistatically on the flux-loss time scale until the central mass-to-flux ratio (dM/dPHI(B))c exceeds the critical value. During quasistatic contraction, the magnetic field lines are essentially ''held in place'' as the neutrals contract through them, and the field strength increases by less than a factor of 2. Despite subsequent dynamic contraction perpendicular to magnetic field lines, thermal pressure continues to balance gravity along field lines, thereby enforcing quasistatic contraction in this direction. We follow the contraction until the central density n(c) increases by a factor of 10(6) (typically from 3 x 10(2) to 3 x 10(8) cm-3). The envelope remains magnetically supported. The results from our parameter study show that decreasing nu(ff, 0) speeds up ambipolar diffusion, shortens the quasistatic phase of contraction, and causes (dM/dPHI(B))c to increase by a greater amount above the critical value. The enhancement of the central magnetic field B(c), however, is not sensitive to the value of nu(ff, 0). A smaller k leads to progressively more rapid ambipolar diffusion as n(c) increases. Reducing mu0 lengthens the duration of the quasistatic phase and results in a larger reduction of a core's magnetic flux, but the dynamic phase of contraction is independent of mu0. The value of (dM/dPHI(B))c in our models is initially 1.9-16.3 times smaller than the critical value, but increases (because of ambipolar diffusion) by a factor of 10-50 by the time n(c) has increased by a factor almost-equal-to 10(6). The value of alpha0 mainly affects the relaxation of clouds from their initial states into the corresponding equilibrium states under flux-freezing, in which alpha is related to mu0 by alpha(c,eq)mu0(2) almost-equal-to constant.
