We have used the Two Micron All Sky Survey and Palomar Observatory Sky Survey photographic plates, digitized by the US Naval Observatory's Precision Measuring Machine program, to derive proper motions over a 100 square degree, spatially complete region centered on the Praesepe open cluster. Proper-motion measurements spanned the magnitude range R~12-19, which covers most of the lower main sequence in Praesepe. The incidence of Hα emission from moderate-resolution (~2600), red spectroscopy of 434 faint candidate members and 126 field control stars demonstrates that ~60%-80% of candidates within 2° of the cluster center are genuine Praesepe members. Spectral index ratios from TiO and CaH absorption features show a well-defined sequence with J magnitude for high-probability candidates, relative to that of field stars. The inferred membership data have allowed us to estimate Praesepe's mass function and general projected structure for stars with mass 0.1-1.0 Msolar. The mass function, when fitted with a power law of the form dN/dm~m-α, rises (α~1.6) from 1.0 to 0.4 Msolar, where it distinctly becomes flat (α~0) down to 0.1 Msolar. The mass function also exhibits a marginal (~1 σ) radial dependence, which, when compared with recent N-body simulations, suggests that the core is depleted of low-mass stars. The outer halo of Praesepe is diffuse and shows a slightly elliptical shape among the highest probability members. We place an upper limit on the surface density of escaped members at 2.2 stars deg-2 outside the limiting radius.