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Doppler measurements from Subaru and Keck have revealed radial velocity variations in the V = 8.15, G0 IV star HD 149026 consistent with a Saturn-mass planet in a 2.8766 day orbit. Photometric observations at Fairborn Observatory have detected three complete transit events with depths of 0.003 mag at the predicted times of conjunction. HD 149026 is now the second-brightest star with a transiting extrasolar planet. The mass of the star, based on interpolation of stellar evolutionary models, is 1.3 ± 0.1 M☉; together with the Doppler amplitude K1 = 43.3 m s-1, we derive a planet mass M sin i = 0.36MJ and orbital radius 0.042 AU. HD 149026 is chromospherically inactive and metal-rich with spectroscopically derived [Fe/H] = +0.36, Teff = 6147 K, log g = 4.26, and v sin i = 6.0 km s-1. Based on Teff and the stellar luminosity of 2.72 L☉, we derive a stellar radius of 1.45 R☉. Modeling of the three photometric transits provides an orbital inclination of 85fdg3 ± 1fdg0 and (including the uncertainty in the stellar radius) a planet radius of (0.725 ± 0.05)RJ. Models for this planet mass and radius suggest the presence of a ~67 M⊕ core composed of elements heavier than hydrogen and helium. This substantial planet core would be difficult to construct by gravitational instability.