Anomalous X-ray pulsars (AXPs) are slowly rotating neutron stars with very bright and highly variable X-ray emission that are believed to be powered by ultra-strong magnetic fields of >1014G, according to the 'magnetar' model1. The radio pulsations that have been observed from more than 1,700 neutron stars with weaker magnetic fields have never been detected from any of the dozen known magnetars. The X-ray pulsar XTE J1810 - 197 was revealed (in 2003) as the first AXP with transient emission when its luminosity increased 100-fold from the quiescent level2; a coincident radio source of unknown origin was detected one year later3. Here we show that XTE J1810 - 197 emits bright, narrow, highly linearly polarized radio pulses, observed at every rotation, thereby establishing that magnetars can be radio pulsars. There is no evidence of radio emission before the 2003 X-ray outburst (unlike ordinary pulsars, which emit radio pulses all the time), and the flux varies from day to day. The flux at all radio frequencies is approximately equal-and at >20 GHz XTE J1810 - 197 is currently the brightest neutron star known. These observations link magnetars to ordinary radio pulsars, rule out alternative accretion models for AXPs, and provide a new window into the coronae of magnetars.