  We present the discovery of the most distant, dynamically relaxed cool core
cluster, SPT-CL J2215-3537 (SPT2215) and its central brightest cluster galaxy
(BCG) at z=1.16. Using new X-ray observations, we demonstrate that SPT2215
harbors a strong cool core, with a central cooling time of 200 Myr (at 10 kpc)
and a maximal intracluster medium cooling rate of 1900+/-400 Msun/yr. This
prodigious cooling may be responsible for fueling extended, star-forming
filaments observed in Hubble Space Telescope imaging. Based on new
spectrophotometric data, we detect bright [O II] emission in the BCG, implying
an unobscured star formation rate (SFR) of 320^{+230}_{-140} Msun/yr. The
detection of a weak radio source (2.0+/-0.8 mJy at 0.8 GHz) suggests ongoing
feedback from an active galactic nucleus (AGN), though the implied jet power is
less than half the cooling luminosity of the hot gas, consistent with cooling
overpowering heating. The extreme cooling and SFR of SPT2215 is rare among
known cool core clusters, and it is even more remarkable that we observe these
at such high redshift, when most clusters are still dynamically disturbed. The
high mass of this cluster, coupled with the fact that it is dynamically relaxed
with a highly-isolated BCG, suggests that it is an exceptionally rare system
that must have formed very rapidly in the early Universe. Combined with the
high SFR, SPT2215 may be a high-z analog of the Phoenix cluster, potentially
providing insight into the limits of AGN feedback and star formation in the
most massive galaxies.
