  We present the first results from a new, deep (200ks) Chandra observation of
the X-ray luminous galaxy cluster surrounding the powerful (L ~10^47 erg/s),
high-redshift (z=1.067), compact-steep-spectrum radio-loud quasar 3C186. The
diffuse X-ray emission from the cluster has a roughly ellipsoidal shape and
extends out to radii of at least ~60 arcsec (~500 kpc). The centroid of the
diffuse X-ray emission is offset by 0.68(+/-0.11) arcsec (5.5+/-0.9 kpc) from
the position of the quasar. We measure a cluster mass within the radius at
which the mean enclosed density is 2500 times the critical density,
r_2500=283(+18/-13)kpc, of 1.02 (+0.21/-0.14)x10^14 M_sun. The gas mass
fraction within this radius is f_gas=0.129(+0.015/-0.016). This value is
consistent with measurements at lower redshifts and implies minimal evolution
in the f_gas(z) relation for hot, massive clusters at 0<z<1.1. The measured
metal abundance of 0.42(+0.08/-0.07) Solar is consistent with the abundance
observed in other massive, high redshift clusters. The spatially-resolved
temperature profile for the cluster shows a drop in temperature, from kT~8 keV
to kT~3 keV, in its central regions that is characteristic of cooling core
clusters. This is the first spectroscopic identification of a cooling core
cluster at z>1. We measure cooling times for the X-ray emitting gas at radii of
50 kpc and 25 kpc of 1.7(+/-0.2)x10^9 years and 7.5(+/-2.6)x 10^8 years, as
well as a nominal cooling rate (in the absence of heating) of
400(+/-190)M_sun/year within the central 100 kpc. In principle, the cooling gas
can supply enough fuel to support the growth of the supermassive black hole and
to power the luminous quasar. The radiative power of the quasar exceeds by a
factor of 10 the kinematic power of the central radio source, suggesting that
radiative heating may be important at intermittent intervals in cluster cores.
