Although the link between long γ-ray bursts (GRBs) and supernovae has been established, hitherto there have been no observations of the beginning of a supernova explosion and its intimate link to a GRB. In particular, we do not know how the jet that defines a γ-ray burst emerges from the star's surface, nor how a GRB progenitor explodes. Here we report observations of the relatively nearby GRB 060218 (ref. 5) and its connection to supernova SN 2006aj (ref. 6). In addition to the classical non-thermal emission, GRB 060218 shows a thermal component in its X-ray spectrum, which cools and shifts into the optical/ultraviolet band as time passes. We interpret these features as arising from the break-out of a shock wave driven by a mildly relativistic shell into the dense wind surrounding the progenitor. We have caught a supernova in the act of exploding, directly observing the shock break-out, which indicates that the GRB progenitor was a Wolf-Rayet star.