We report precise radial velocity measurements of the K giant iota Dra (HD 137759, HR 5744, HIP 75458), carried out at Lick Observatory, which reveal the presence of a substellar companion orbiting the primary star. A Keplerian fit to the data yields an orbital period of about 536 days and an eccentricity of 0.70. Assuming a mass of 1.05 M-circle dot for iota Dra, the mass function implies a minimum companion mass m(2) sin i of 8.9 M-J, making it a planet candidate. The corresponding semimajor axis is 1.3 AU. The nondetection of the orbital motion by Hipparcos allows us to place an upper limit of 45 M-J on the companion mass, establishing the substellar nature of the object. We estimate that transits in this system could occur already for inclinations as low as 81.5degrees, as a result of the large diameter of the giant star. The companion to iota Dra is the first brown dwarf or planet found to orbit a giant rather than a main-sequence star.