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We report the detection of the cool, Jovian-mass planet MOA-2007-BLG-400Lb. The planet was detected in a high-magnification microlensing event (with peak magnification A_max = 628) in which the primary lens transited the source, resulting in a dramatic smoothing of the peak of the event. The angular extent of the region of perturbation due to the planet is significantly smaller than the angular size of the source, and as a result the planetary signature is also smoothed out by the finite source size. Thus the deviation from a single-lens fit is broad and relatively weak (~ few percent). Nevertheless, we demonstrate that the planetary nature of the deviation can be unambiguously ascertained from the gross features of the residuals, and detailed analysis yields a fairly precise planet/star mass ratio of q = 0.0026+/-0.0004, in accord with the large significance (\Delta\chi^2=1070) of the detection. The planet/star projected separation is subject to a strong close/wide degeneracy, leading to two indistinguishable solutions that differ in separation by a factor of ~8.5. Upper limits on flux from the lens constrain its mass to be M < 0.75 M_Sun (assuming it is a main-sequence star). A Bayesian analysis that includes all available observational constraints indicates a primary in the Galactic bulge with a mass of ~0.2-0.5 M_Sun and thus a planet mass of ~ 0.5-1.3 M_Jupiter. The separation and equilibrium temperature are ~0.6-1.1AU (~5.3-9.7AU) and ~103K (~34K) for the close (wide) solution. If the primary is a main-sequence star, follow-up observations would enable the detection of its light and so a measurement of its mass and distance., Comment: 30 pages, 6 figures, Submitted to ApJ