Pulsars are thought to eject electron-positron winds that energize the surrounding environment, with the formation of a pulsar wind nebula(1). The pulsar wind originates close to the light cylinder, the surface at which the pulsar co-rotation velocity equals the speed of light, and carries away much of the rotational energy lost by the pulsar. Initially the wind is dominated by electromagnetic energy (Poynting flux) but later this is converted to the kinetic energy of bulk motion(2). It is unclear exactly where this takes place and to what speed the wind is accelerated. Although some preferred models imply a gradual acceleration over the entire distance from the magnetosphere to the point at which the wind terminates(3,4), a rapid acceleration close to the light cylinder cannot be excluded(5,6). Here we report that the recent observations of pulsed, very high-energy gamma-ray emission from the Crab pulsar(7-9) are explained by the presence of a cold (in the sense of the low energy of the electrons in the frame of the moving plasma) ultrarelativistic wind dominated by kinetic energy. The conversion of the Poynting flux to kinetic energy should take place abruptly in the narrow cylindrical zone of radius between 20 and 50 light-cylinder radii centred on the axis of rotation of the pulsar, and should accelerate the wind to a Lorentz factor of (0.5-1.0) x 10(6). Although the ultrarelativistic nature of the wind does support the general model of pulsars, the requirement of the very high acceleration of the wind in a narrow zone not far from the light cylinder challenges current models.