In the cores of some clusters of galaxies the hot intracluster plasma is dense enough that it should cool radiatively in the cluster's lifetime, leading to continuous `cooling flows' of gas sinking towards the cluster centre, yet no such cooling flow has been observed. The low observed star-formation rates and cool gas masses for these `cool-core' clusters suggest that much of the cooling must be offset by feedback to prevent the formation of a runaway cooling flow. Here we report X-ray, optical and infrared observations of the galaxy cluster SPT-CLJ2344-4243 (ref. 11) at redshift z = 0.596. These observations reveal an exceptionally luminous (8.2 × 1045 erg s-1) galaxy cluster that hosts an extremely strong cooling flow (around 3,820 solar masses a year). Further, the central galaxy in this cluster appears to be experiencing a massive starburst (formation of around 740 solar masses a year), which suggests that the feedback source responsible for preventing runaway cooling in nearby cool-core clusters may not yet be fully established in SPT-CLJ2344-4243. This large star-formation rate implies that a significant fraction of the stars in the central galaxy of this cluster may form through accretion of the intracluster medium, rather than (as is currently thought) assembling entirely via mergers.