Acceleration of particles by intense laser-plasma interactions represents a rapidly evolving field of interest, as highlighted by the recent demonstration of laser-driven relativistic beams of monoenergetic electrons. Ultrahigh-intensity lasers can produce accelerating fields of 10 TV m(-1) (1 TV = 10(12) V), surpassing those in conventional accelerators by six orders of magnitude. Laser-driven ions with energies of several MeV per nucleon have also been produced. Such ion beams exhibit unprecedented characteristics--short pulse lengths, high currents and low transverse emittance--but their exponential energy spectra have almost 100% energy spread. This large energy spread, which is a consequence of the experimental conditions used to date, remains the biggest impediment to the wider use of this technology. Here we report the production of quasi-monoenergetic laser-driven C5+ ions with a vastly reduced energy spread of 17%. The ions have a mean energy of 3 MeV per nucleon (full-width at half-maximum approximately 0.5 MeV per nucleon) and a longitudinal emittance of less than 2 x 10(-6) eV s for pulse durations shorter than 1 ps. Such laser-driven, high-current, quasi-monoenergetic ion sources may enable significant advances in the development of compact MeV ion accelerators, new diagnostics, medical physics, inertial confinement fusion and fast ignition.