We report on the detection of a planetary companion in orbit around the primary star of the binary system γ Cephei. High-precision radial velocity measurements using four independent data sets spanning the time interval 1981-2002 reveal long-lived residual radial velocity variations superposed on the binary orbit that are coherent in phase and amplitude with a period or 2.48 yr (906 days) and a semiamplitude of 27.5 m s^-1. We performed a careful analysis of our Ca II H and KS-index measurements, spectral line bisectors, andHipparcos photometry. We found no significant variations in these quantities with the 906 day period. We also reanalyzed the Ca II λ8662 measurements of Walker et al., which showed possible periodic variations with the "planet" period when first published. This analysis shows that periodic Ca II equivalent width variations were only present during 1986.5-1992 and absent during 1981-1986.5. Furthermore, a refined period for the Ca II λ8662 variations is 2.14 yr, significantly less than the residual radial velocity period. The most likely explanation of the residual radial velocity variations is a planetary-mass companion withM sini = 1.7M_J and an orbital semimajor axis ofa₂ = 2.13 AU. This supports the planet hypothesis for the residual radial velocity variations for γ Cep first suggested by Walker et al. With an estimated binary orbital period of 57 yr, γ Cep is the shortest period binary system in which an extrasolar planet has been found. This system may provide insights into the relationship between planetary and binary star formation.