It is thought that neutron stars in low-mass binary systems can accrete matter and angular momentum from the companion star and be spun-up to millisecond rotational periods(1-3). During the accretion stage, the system is called a low-mass X-ray binary, and bright X-ray emission is observed. When the rate of mass transfer decreases in the later evolutionary stages, these binaries host a radio millisecond pulsar(4,5) whose emission is powered by the neutron star's rotating magnetic field(6). This evolutionary model is supported by the detection of millisecond X-ray pulsations from several accreting neutron stars(7,8) and also by the evidence for a past accretion disc in a rotation-powered millisecond pulsar(9). It has been proposed that a rotation-powered pulsar may temporarily switch on(10-12) during periods of low mass inflow(13) in some such systems. Only indirect evidence for this transition has hitherto been observed(14-18). Here we report observations of accretion-powered, millisecond X-ray pulsations from a neutron star previously seen as a rotation-powered radio pulsar. Within a few days after a month-long X-ray outburst, radio pulses were again detected. This not only shows the evolutionary link between accretion and rotation-powered millisecond pulsars, but also that some systems can swing between the two states on very short timescales.