We report precise radial velocity measurements of the K giant ɩ 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 M8 for ɩ Dra, the mass function implies a minimum companion mass m2 sin i of 8.9 MJ, 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 MJ on the companion mass, establishing the sub-stellar nature of the object. We estimate that transits in this system could occur already for inclinations as low as 81.5o, as a result of the large diameter of the giant star. The companion to ɩ Dra is the first brown dwarf or planet found to orbit a giant rather than a main-sequence star.