Considerable uncertainties remain about the nature of warm, AGN-driven outflows and their impact on the evolution of galaxies. This is because the outflows are often unresolved in ground-based observations. As part of a project to study the AGN outflows in some of the most rapidly evolving galaxies in the local Universe, here we present Hubble Space Telescope (HST)/Space Telescope Imaging Spectrograph (STIS) observations of F14394+5332E that resolve the sub-kpc warm outflow for the first time in an ultra-luminous infrared galaxy. The observations reveal a compact, high-ionization outflow region (rmax ˜ 0.9 kpc) set in a more extensive (rmax ˜ 1.4 kpc) halo that is kinematically quiescent and has a lower ionization state. A large line width (600 < FWHM < 1500 km s-1) is measured throughout the outflow region, and the outflowing gas shows a steep velocity gradient with radius, with the magnitude of the blueshifted velocities increasing from ˜500 to 1800 km s-1 from the inner to the outer part of the outflow. We interpret the observations in terms of the local acceleration, and hydrodynamic destruction, of dense clouds as they are swept up in a hot, low-density wind driven by the AGN. We discuss the implications for measuring the mass outflow rates and kinetic powers for the AGN-driven outflows in such objects.