Transfer of angular momentum from a spin-polarized current to a ferromagnet provides an efficient means to control the magnetization dynamics of nanomagnets. A peculiar consequence of this spin torque, the ability to induce persistent oscillations in a nanomagnet by applying a d.c. current, has previously been reported only for spatially uniform nanomagnets. Here, we demonstrate that a quintessentially non-uniform magnetic structure, a magnetic vortex, isolated within a nanoscale spin-valve structure, can be excited into persistent microwave-frequency oscillations by a spin-polarized d.c. current. Comparison with micromagnetic simulations leads to identification of the oscillations with a precession of the vortex core. The oscillations, which can be obtained in essentially zero magnetic field, exhibit linewidths that can be narrower than 300kHz at 1.1GHz, making these highly compact spin-torque vortex-oscillator devices potential candidates for microwave signal-processing applications, and a powerful new tool for fundamental studies of vortex dynamics in magnetic nanostructures.