We combine all available information to constrain the nature of OGLE-2005-BLG-071Lb, the second planet discovered by microlensing and the first in a high-magnification event. These include photometric and astrometric measurements from the Hubble Space Telescope, as well as constraints from higher order effects extracted from the ground-based light curve, such as microlens parallax, planetary orbital motion, and finite-source effects. Our primary analysis leads to the conclusion that the host of Jovian planet OGLE-2005-BLG-071Lb is an M dwarf in the foreground disk with mass M = 0.46 ± 0.04 M, distance Dl = 3.2 ± 0.4 kpc, and thick-disk kinematics vLSR ~ 103 km s−1. From the best-fit model, the planet has mass Mp = 3.8 ± 0.4 MJupiter, lies at a projected separation r = 3.6 ± 0.2AU from its host, and so has an equilibrium temperature of T ~ 55 K, that is, similar to Neptune. A degenerate model gives similar planetary mass Mp = 3.4 ± 0.4 MJupiter with a smaller projected separation, r = 2.1 ± 0.1AU, and higher equilibrium temperature, T ~ 71 K. These results from the primary analysis suggest that OGLE-2005-BLG-071Lb is likely to be the most massive planet yet discovered that is hosted by an M dwarf. However, the formation of such high-mass planetary companions in the outer regions of M dwarf planetary systems is predicted to be unlikely within the core-accretion scenario. There are a number of caveats to this primary analysis, which assumes (based on real but limited evidence) that the unlensed light coincident with the source is actually due to the lens, that is, the planetary host. However, these caveats could mostly be resolved by a single astrometric measurement a few years after the event.