[en] The association of a supernova with GRB030329 strongly supports the `collapsar' model of γ-ray bursts, where a relativistic jet forms after the progenitor star collapses. Such jets cannot be spatially resolved because γ-ray bursts lie at cosmological distances; their existence is instead inferred from `breaks' in the light curves of the afterglows, and from the theoretical desire to reduce the estimated total energy of the burst by proposing that most of it comes out in narrow beams. Temporal evolution of the polarization of the afterglows may provide independent evidence for the jet structure of the relativistic outflow. Small-level polarization (~1-3 per cent) has been reported for a few bursts, but its temporal evolution has yet to be established. Here we report polarimetric observations of the afterglow of GRB030329. We establish the polarization light curve, detect sustained polarization at the per cent level, and find significant variability. The data imply that the afterglow magnetic field has a small coherence length and is mostly random, probably generated by turbulence, in contrast with the picture arising from the high polarization detected in the prompt γ-rays from GRB021206 (ref. 18).