We analyze the Type II plateau supernovae (SNe IIP) SN 2005cs and SN 2006bp with the non-LTE model atmosphere code CMFGEN. We fit 13 spectra in the first month for SN 2005cs and 18 for SN 2006bp. Swift ultraviolet photometry and ground-based optical photometry calibrate each spectrum. Our analysis shows that both objects were discovered less than 3 days after they exploded, making these the earliest SN IIP spectra ever studied. They reveal broad and very weak lines from highly ionized fast ejecta with an extremely steep density profile. We identify He II λ4686 emission in the SN 2006bp ejecta. Days later, the spectra resemble the prototypical Type IIP SN 1999em, which had a supergiant-like photospheric composition. Despite the association of SN 2005cs with possible X-ray emission, the emergent UV and optical light comes from the photosphere, not from circumstellar emission. We surmise that the very steep density falloff we infer at early times may be a fossil of the combined actions of the shock wave passage and radiation driving at shock breakout. Based on tailored CMFGEN models, the direct fitting technique and the expanding photosphere method both yield distances and explosion times that agree within a few percent. We derive a distance to NGC 5194, the host of SN 2005cs, of 8.9 ± 0.5 Mpc and 17.5 ± 0.8 Mpc for SN 2006bp in NGC 3953. The luminosity of SN 2006bp is 1.5 times that of SN 1999em and 6 times that of SN 2005cs. Reliable distances to SNe IIP that do not depend on a small range in luminosity provide an independent route to the Hubble constant and improved constraints on other cosmological parameters.