  We present an idealized, spherical model of the evolution of a magnetized
molecular cloud due to ambipolar diffusion. This model allows us to follow the
quasi-static evolution of the cloud's core prior to collapse and the subsequent
evolution of the remaining envelope. By neglecting the thermal pressure
gradients in comparison with magnetic stresses and by assuming that the ion
velocity is small compared with the neutral velocity, we are able to find exact
analytic solutions to the MHD equations. We show that, in the case of a
centrally condensed cloud, a core of finite mass collapses into the origin
leaving behind a quasi-static envelope, whereas initially homogeneous clouds
never develop any structure in the absence of thermal stresses, and collapse as
a whole. Prior to the collapse of the core, the cloud's evolution is
characterized by two phases: a long, quasi-static phase where the relevant
timescale is the ambipolar diffusion time (treated in this paper), and a short,
dynamical phase where the characteristic timescale is the free-fall time. The
collapse of the core is an "outside-in" collapse. The quasi-static evolution
terminates when the cloud becomes magnetically supercritical; thereafter its
evolution is dynamical, and a singularity develops at the origin-a protostar.
After the initial formation of the protostar, the outer envelope continues to
evolve quasi-statically, while the region of dynamical infall grows with
time-an "inside-out" collapse. We use our solution to estimate the magnetic
flux trapped in the collapsing core and the mass accretion rate onto the newly
formed protostar. Our results agree, within factors of order unity, with the
numerical results of Fiedler & Mouschovias (1992) for the physical quantities
in the midplane of
