We report on rotating an optically trapped silica nanoparticle in vacuum by transferring spin angular momentum of light to the particle's mechanical angular momentum. At sufficiently low damping, realized at pressures below 10^{-5} mbar, we observe rotation frequencies of single 100 nm particles exceeding 1 GHz. We find that the steady-state rotation frequency scales linearly with the optical trapping power and inversely with pressure, consistent with theoretical considerations based on conservation of angular momentum. Rapidly changing the polarization of the trapping light allows us to extract the pressure-dependent response time of the particle's rotational degree of freedom.