  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.
