We present an extensive ALMA spectroscopic follow-up programme of the $z,{=},4.3$ structure SPT2349$-$56, one of the most actively star-forming proto-cluster cores known, to identify additional members using their [C{sc ii}] 158,$mu$m and mbox{CO(4--3)} lines. In addition to robustly detecting the 14 previously published galaxies in this structure, we identify a further 15 associated galaxies at $z,{=},4.3$, resolving 55$,{pm},$5,per cent of the 870-$mu$m flux density at 0.5,arcsec resolution compared to 21,arcsec single-dish data. These galaxies are distributed into a central core containing 23 galaxies extending out to 300,kpc in diameter, and a northern extension, offset from the core by 400,kpc, containing three galaxies. We discovered three additional galaxies in a red {it Herschel/}-SPIRE source 1.5,Mpc from the main structure, suggesting the existence of many other sources at the same redshift as SPT2349$-$56 that are not yet detected in the limited coverage of our data. An analysis of the velocity distribution of the central galaxies indicates that this region may be virialized with a mass of (9$pm$5)$,{times},$10$^{12}$,M$_{odot}$, while the two offset galaxy groups are about 30 and 60,per cent less massive and show significant velocity offsets from the central group. We calculate the [C{sc ii}] and far-infrared number counts, and find evidence for a break in the [C{sc ii}] luminosity function. We estimate the average SFR density within the region of SPT2349$-$56 containing single-dish emission (a proper diametre of 720,kpc), assuming spherical symmetry, to be roughly 4$,{times},10^4$,M$_{odot}$,yr$^{-1}$,Mpc$^{-3}$; this may be an order of magnitude greater than the most extreme examples seen in simulations.
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