Massive galaxy clusters are now found as early as 3 billion years after the Big Bang, containing stars that formed at even earlier epochs. The high-redshift progenitors of these galaxy clusters, termed protoclusters, are identified in cosmological simulations with the highest dark matter overdensities. While their observational signatures are less well defined compared to virialized clusters with a substantial hot intra-cluster medium (ICM), protoclusters are expected to contain extremely massive galaxies that can be observed as luminous starbursts. Recent claimed detections of protoclusters hosting such starbursts do not support the kind of rapid cluster core formation expected in simulations because these structures contain only a handful of starbursting galaxies spread throughout a broad structure, with poor evidence for eventual collapse into a protocluster. Here we report that the source SPT2349-56 consists of at least 14 gas-rich galaxies all lying at z = 4.31 based on sensitive observations of carbon monoxide and ionized carbon. We demonstrate that each of these galaxies is forming stars between 50 and 1000 times faster than our own Milky Way, and all are located within a projected region only $sim$ 130 kiloparsecs in diameter. This galaxy surface density is more than 10 times the average blank field value (integrated over all redshifts) and $>$1000 times the average field volume density. The velocity dispersion ($sim$ 410 km s$^{-1}$) of these galaxies and enormous gas and star formation densities suggest that this system represents a galaxy cluster core at an advanced stage of formation when the Universe was only 1.4 billion years old. A comparison with other known protoclusters at high redshifts shows that SPT2349-56 is a uniquely massive and dense system that could be building one of the most massive structures in the Universe today.
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