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Submillimeter bright galaxies in the early Universe are vigorously forming stars at ~1000 times higher rate than the Milky Way. A large fraction of stars is formed in the central 1 kiloparsec region, that is comparable in size to massive, quiescent galaxies found at the peak of the cosmic star formation history, and eventually the core of giant elliptical galaxies in the present-day Universe. However, the physical and kinematic properties inside a compact starburst core are poorly understood because dissecting it requires angular resolution even higher than the Hubble Space Telescope can offer. Here we report 550 parsec-resolution observations of gas and dust in the brightest unlensed submillimeter galaxy at z=4.3. We map out for the first time the spatial and kinematic structure of molecular gas inside the heavily dust-obscured core. The gas distribution is clumpy while the underlying disk is rotation-supported. Exploiting the high-quality map of molecular gas mass surface density, we find a strong evidence that the starburst disk is gravitationally unstable, implying that the self-gravity of gas overcomes the differential rotation and the internal pressure by stellar radiation feedback. The observed molecular gas would be consumed by star formation in a timescale of 100 million years, that is comparable to those in merging starburst galaxies. Our results suggest that the most extreme starburst in the early Universe originates from efficient star formation due to a gravitational instability in the central 2 kpc region., Published in Nature on August 30 2018 (submitted version)