Powerful winds driven by active galactic nuclei (AGN) are often invoked to play a fundamental role in the evolution of both supermassive black holes (SMBHs) and their host galaxies, quenching star formation and explaining the tight SMBH-galaxy relations. Recent observations of large-scale molecular outflows in ultra-luminous infrared galaxies (ULIRGs) have provided the evidence to support these studies, as they directly trace the gas out of which stars form. Theoretical models suggest an origin of these outflows as energy-conserving flows driven by fast AGN accretion disk winds. Previous claims of a connection between large-scale molecular outflows and AGN activity in ULIRGs were incomplete because they were lacking the detection of the putative inner wind. Conversely, studies of powerful AGN accretion disk winds to date have focused only on X-ray observations of local Seyferts and a few higher redshift quasars. Here we show the clear detection of a powerful AGN accretion disk wind with a mildly relativistic velocity of 0.25c in the X-ray spectrum of IRAS F11119+3257, a nearby (z = 0.189) optically classified type 1 ULIRG hosting a powerful molecular outflow. The AGN is responsible for ~80% of the emission, with a quasar-like luminosity of L_AGN = 1.5x10^46 erg/s. The energetics of these winds are consistent with the energy-conserving mechanism, which is the basis of the quasar mode feedback in AGN lacking powerful radio jets.