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We report precise Doppler measurements of seven subgiants from Keck Observatory. All seven stars show variability in their radial velocities consistent with planet-mass companions in Keplerian orbits. The host stars have masses ranging from 1.1 ≤ M⋆/M⊙ ≤ 1.9, radii 3.4 ≤ R⋆/R⊙ ≤ 6.1, and metallicities -0.21 ≤ [Fe/H] ≤ +0.26. The planets are all more massive than Jupiter (MP sin i > 1 MJup) and have semimajor axes a > 1 AU. We present millimagnitude photometry from the T3 0.4 m APT at Fairborn Observatory for five of the targets. Our monitoring shows these stars to be photometrically stable, further strengthening the interpretation of the observed radial velocity variability. The orbital characteristics of the planets thus far discovered around former A-type stars are very different from the properties of planets around dwarf stars of spectral type F, G, and K, and suggests that the formation and migration of planets is a sensitive function of stellar mass. Three of the planetary systems show evidence of long-term, linear trends indicative of additional distant companions. These trends, together with the high planet masses and increased occurrence rate, indicate that A-type stars are very promising targets for direct-imaging surveys.