Galaxies in the early Universe that are bright at submillimetre wavelengths (submillimetre-bright galaxies) are forming stars at a rate roughly 1,000 times higher than the Milky Way. A large fraction of the new stars form in the central kiloparsec of the galaxy(1-3), a region that is comparable in size to the massive, quiescent galaxies found at the peak of cosmic star-formation history' and the cores of present-day giant elliptical galaxies. The physical and kinematic properties inside these compact starburst cores are poorly understood because probing them at relevant spatial scales requires extremely high angular resolution. Here we report observations with a linear resolution of 550 parsecs of gas and dust in an unlensed, submillimetre-bright galaxy at a redshift of z = 4.3, when the Universe was less than two billion years old. We resolve the spatial and kinematic structure of the molecular gas inside the heavily dust-obscured core and show that the underlying gas disk is clumpy and rotationally supported (that is, its rotation velocity is larger than the velocity dispersion). Our analysis of the molecular gas mass per unit area suggests that the starburst disk is gravitationally unstable, which implies that the self-gravity of the gas is stronger than the differential rotation of the disk and the internal pressure due to stellar-radiation feedback. As a result of the gravitational instability in the disk, the molecular gas would be consumed by star formation on a timescale of 100 million years, which is comparable to gas depletion times in merging starburst galaxies(5).