We present Hubble Space Telescope Wide Field Camera 3 observations of the core of the Phoenix Cluster ( SPT-CLJ2344-4243 ) in five broadband filters spanning rest-frame 1000–5500Å . These observations reveal complex , filamentary blue emission , extending for > 40 kpc from the brightest cluster galaxy . We observe an underlying , diffuse population of old stars , following an r ^ { 1 / 4 } distribution , confirming that this system is somewhat relaxed . The spectral energy distribution in the inner part of the galaxy , as well as along the extended filaments , is a smooth continuum and is consistent with that of a star-forming galaxy , suggesting that the extended , filamentary emission is not due to the central AGN , either from a large-scale ionized outflow or scattered polarized UV emission , but rather a massive population of young stars . We estimate an extinction-corrected star formation rate of 798 \pm 42 M _ { \odot } yr ^ { -1 } , consistent with our earlier work based on low spatial resolution ultraviolet , optical , and infrared imaging . The lack of tidal features and multiple bulges , combine with the need for an exceptionally massive ( > 10 ^ { 11 } M _ { \odot } ) cold gas reservoir , suggest that this star formation is not the result of a merger of gas-rich galaxies . Instead , we propose that the high X-ray cooling rate of \sim 2700 M _ { \odot } yr ^ { -1 } is the origin of the cold gas reservoir . The combination of such a high cooling rate and the relatively weak radio source in the cluster core suggests that feedback has been unable to halt cooling in this system , leading to this tremendous burst of star formation .