We report FUSE far-UV spectroscopy of the prototypical dwarf starburst galaxy NGC 1705. These data allow us for the first time to directly probe the coronal-phase (T = a few times 105 K) gas that may dominate the radiative cooling of the supernova-heated interstellar medium (ISM) and thereby determine the dynamical evolution of the starburst-driven outflows in dwarf galaxies. We detect a broad (∼100 km s-1 FWHM) and blueshifted (Δv = 77 km s-1) O VT λ032 absorption line arising in the previously known galactic outflow. The mass and kinetic energy in the outflow we detect is dominated by the warm (T ∼ 104 K) photoionized gas which is also seen through its optical line emission. The kinematics of this warm gas are compatible with a simple model of the adiabatic expansion of a superbubble driven by the collective effect of the kinetic energy supplied by supernovae in the starburst. However, the observed properties of the O VI absorption in NGC 1705 are not consistent with the simple superbubble model, in which the O VI would arise in a conductive interface inside the superbubble's outer shell. The relative outflow speed of the O VI is too high and the observed column density (log NOVI = 14.3) is much too large. We argue that the superbubble has begun to blow out of the ISM of NGC 1705. During this blowout phase the superbubble shell accelerates and fragments. The resulting hydrodynamical interaction as hot outrushing gas flows between the cool shell fragments will create intermediate-temperature coronal gas that can produce the observed O VI absorption. For the observed flow speed of ∼102 km s-1, the observed O VI column density is just what is expected for gas that has been heated and which then cools radiatively. Assuming that the coronal-phase gas is in rough pressure balance with the warm photoionized gas, we estimate a cooling rate of order ∼0.1 M⊙ yr-1 and ∼1039 ergs s-1 in the coronal gas. The latter represents less than 10% of the supernova heating rate. Independent of the assumed pressure, the lack of observed redshifted O VI emission from the back side of the outflow leads to upper limits on the cooling rate of ≤20% of the supernova heating rate. Since the X-ray luminosity of NGC 1705 is negligible, we conclude that radiative losses are insignificant in the outflow. The outflow should therefore be able to fully blow out of the ISM of NGC 1705 and vent its metals and kinetic energy. This process has potentially important implications for the evolution of dwarf galaxies and the intergalactic medium.