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) sub-halos, in light of the dozen new low luminosity satellites found in SDSS imaging data and our recent calibration of the SDSS satellite detection efficiency, which implies a total population far larger than these dozen discoveries. We combine a dynamical model for the CDM sub-halo population with simple, physically motivated prescriptions for assigning stellar content to each sub-halo, then apply observational selection effects and compare to the current observational census. As expected, models in which the stellar mass is a constant fraction F(Omega_b/Omega_m) of the sub-halo 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_c <~ 35 km/s can reproduce the observed satellite counts for -15 < M_V < 0, with F ~ 10^{-3}. Successful models also require strong suppression of star formation BEFORE reionization in halos with v_c <~ 10 km/s; models without pre-reionization suppression predict far too many satellites with -5 < M_V < 0. Our models also reproduce the observed stellar velocity dispersions ~ 5-10 km/s of the SDSS dwarfs given the observed sizes of their stellar distributions, and model satellites have M(<300 pc) ~ 10^7 M_sun as observed even though their present day total halo masses span more than two orders of magnitude. 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.
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