As part of a survey for radio pulsars with the Parkes 64 m telescope, we have discovered PSR J1622-4950, a pulsar with a 4.3 s rotation period. Follow-up observations show that the pulsar has the highest inferred surface magnetic field of the known radio pulsars (B ∼ 3 × 1014 G), and it exhibits significant timing noise and appears to have an inverted spectrum. Unlike the vast majority of the known pulsar population, PSR J1622-4950 appears to switch off for many hundreds of days and even in its on-state exhibits extreme variability in its flux density. Furthermore, the integrated pulse profile changes shape with epoch. All of these properties are remarkably similar to the only two magnetars previously known to emit radio pulsations. The position of PSR J1622-4950 is coincident with an X-ray source that, unlike the other radio pulsating magnetars, was found to be in quiescence. We conclude that our newly discovered pulsar is a magnetar - the first to be discovered via its radio emission. © 2010. The American Astronomical Society. All rights reserved.