We revisit the well known discrepancy between the observed number of Milky Way (MW) dwarf satellite companions and the predicted population of cold dark matter (CDM) subhalos, in light of the dozen new low-luminosity satellites found in imaging data from the Sloan Digital Sky Survey (SDSS) and our recent calibration of the SDSS satellite detection efficiency, which implies a total satellite population far larger than these dozen discoveries. We combine a detailed dynamical model for the CDM subhalo population with simple, physically motivated prescriptions for assigning a stellar content to each subhalo, then apply observational selection effects and compare to the current observational census. Reconciling the observed satellite population with CDM predictions still requires strong mass-dependent suppression of star formation in low-mass subhalos: models in which the stellar mass is a constant fraction F *(Ω b /Ω m ) of the subhalo mass M sat at the time it becomes a satellite fail for any choice of F *. However, previously advocated models that invoke suppression of gas accretion after reionization in halos with circular velocity V circ ≤ V crit ≈ 35 km s–1 can reproduce the observed satellite counts for –15 ≤ MV ≤ 0. Successful models require F * ≈ 10–3 in halos with V circ>V crit and strong suppression of star formation before reionization in halos with V circ lesssim 10 km s–1; models without pre-reionization suppression predict far too many satellites with –5 ≤ MV ≤ 0. In this successful model, the dominant fraction of stars formed after reionization at all luminosities. Models that match the satellite luminosity distribution also match the observed heliocentric radius distribution, and they reproduce the observed characteristic stellar velocity dispersion σ* ≈ 5-10 km s–1 of the SDSS dwarfs given the observed sizes (~50-200 pc) of their stellar distributions. The model satellites have M(<300 pc) ~ 107 M ☉ as observed even though their present-day total halo masses span more than two orders of magnitude; the constancy of central masses mainly reflects the profiles of CDM halos. Our modeling shows that natural physical mechanisms acting within the CDM framework can quantitatively explain the properties of the MW satellite population as it is presently known, thus providing a convincing solution to the "missing satellite" problem.