Pulsar wind nebulae are formed when outflows of relativistic electrons and positrons hit the surrounding supernova remnant or interstellar medium at a shock front. The Vela pulsar wind nebula is powered by a young pulsar (B0833-45, aged 11,000 years)1 and located inside an extended structure called Vela X, which is itself inside the supernova remnant2. Previous X-ray observations revealed two prominent arcs that are bisected by a jet and counter jet3,4. Radio maps have shown high linear polarization of 60% in the outer regions of the nebula5. Here we report an X-ray observation of the inner part of the nebula, where polarization can exceed 60% at the leading edge—approaching the theoretical limit of what can be produced by synchrotron emission. We infer that, in contrast with the case of the supernova remnant, the electrons in the pulsar wind nebula are accelerated with little or no turbulence in a highly uniform magnetic field.