The host galaxy of the quasar SDSS J114816.64+525150.3 (at redshift z = 6.42, when the Universe was less than a billion years old) has an infrared luminosity of 2.2 x 10(13) times that of the Sun, presumably significantly powered by a massive burst of star formation. In local examples of extremely luminous galaxies, such as Arp 220, the burst of star formation is concentrated in a relatively small central region of <100 pc radius. It is not known on which scales stars are forming in active galaxies in the early Universe, at a time when they are probably undergoing their initial burst of star formation. We do know that at some early time, structures comparable to the spheroidal bulge of the Milky Way must have formed. Here we report a spatially resolved image of [C ii] emission of the host galaxy of J114816.64+525150.3 that demonstrates that its star-forming gas is distributed over a radius of about 750 pc around the centre. The surface density of the star formation rate averaged over this region is approximately 1,000 year(-1) kpc(-2). This surface density is comparable to the peak in Arp 220, although about two orders of magnitude larger in area. This vigorous star-forming event is likely to give rise to a massive spheroidal component in this system.