Over the past decade, long-duration gamma-ray bursts (GRBs)--including the subclass of X-ray flashes (XRFs)--have been revealed to be a rare variety of type Ibc supernova. Although all these events result from the death of massive stars, the electromagnetic luminosities of GRBs and XRFs exceed those of ordinary type Ibc supernovae by many orders of magnitude. The essential physical process that causes a dying star to produce a GRB or XRF, and not just a supernova, is still unknown. Here we report radio and X-ray observations of XRF 060218 (associated with supernova SN 2006aj), the second-nearest GRB identified until now. We show that this event is a hundred times less energetic but ten times more common than cosmological GRBs. Moreover, it is distinguished from ordinary type Ibc supernovae by the presence of 10(48) erg coupled to mildly relativistic ejecta, along with a central engine (an accretion-fed, rapidly rotating compact source) that produces X-rays for weeks after the explosion. This suggests that the production of relativistic ejecta is the key physical distinction between GRBs or XRFs and ordinary supernovae, while the nature of the central engine (black hole or magnetar) may distinguish typical bursts from low-luminosity, spherical events like XRF 060218.