  We present the best sensitivity and angular resolution maps of the molecular
disk and outflow of Mrk 231, as traced by CO observations obtained with
IRAM/PdBI, and we analyze archival Chandra and NuSTAR observations. We
constrain the physical properties of both the molecular disk and outflow, the
presence of a highly-ionized ultra-fast nuclear wind, and their connection. The
molecular outflow has a size of ~1 kpc, and extends in all directions around
the nucleus, being more prominent along the south-west to north-east direction,
suggesting a wide-angle biconical geometry. The maximum projected velocity of
the outflow is nearly constant out to ~1 kpc, thus implying that the density of
the outflowing material decreases from the nucleus outwards as $r^{-2}$. This
suggests that either a large part of the gas leaves the flow during its
expansion or that the bulk of the outflow has not yet reached out to ~1 kpc,
thus implying a limit on its age of ~1 Myr. We find $\dot M_{OF}=[ 500-1000]~
M_{\odot}~yr^{-1}$ and $\dot E_{kin,OF}=[7-10]\times 10^{43}$ erg s$^{-1}$.
Remarkably, our analysis of the X-ray data reveals a nuclear ultra-fast outflow
(UFO) with velocity -20000 km s$^{-1}$, $\dot M_{UFO}=[0.3- 2.1] ~M_\odot
yr^{-1}$, and momentum load $\dot P_{UFO}/\dot P_{rad}=[0.2-1.6]$.We find $\dot
E_{kin,UFO}\sim \dot E_{kin,OF}$ as predicted for outflows undergoing an energy
conserving expansion. This suggests that most of the UFO kinetic energy is
transferred to mechanical energy of the kpc-scale outflow, strongly supporting
that the energy released during accretion of matter onto super-massive black
holes is the ultimate driver of giant massive outflows. We estimate a momentum
boost $\dot P_{OF}/\dot P_{UFO}\approx [30-60]$. The ratios $\dot E_{kin,
UFO}/L_{bol,AGN} =[ 1-5]\%$ and $\dot E_{kin,OF}/L_{bol,AGN} = [1-3]\%$ agree
with the requirements of the most popular models of AGN feedback.
