We show that the starburst-driven outflow in the peculiar galaxy NGC 2782 forms a well-defined collimated bubble that has an extent of similar to 1 kpc and a closed shell at its edge, as seen in H alpha, [O III], and the 5 GHz radio continuum. The shell coincides with the maximum in intensity and line widths of [O III] lines in a blueshifted emission nebula that was previously detected via optical spectroscopy by Beer et al. in 1992. Such a remarkable outflow morphology has not been observed to date in ally other starburst galaxy of comparable luminosity. The radio continuum map reveals a second bubble of similar size on the opposite side of the nucleus, forming a striking double-bubble outflow morphology. We argue from the morphology and short timescale (similar to 4 x 10(6) yr) of the outflow that it is dynamically younger than freely expanding outflows seen in other galaxies that harbor circumnuclear starbursts of comparable luminosity, e.g., M82. We suggest that the outflow in NGC 2782 is in the early stage where thermal instabilities have not yet completely ruptured the outflow bubble. We present evidence that the outflow is driving warm and hot ionized gas, and possibly cold molecular gas, out of the central kiloparsec of the galaxy. We estimate the contribution of the hot, warm, and cold phases of the interstellar medium to the energetics of the outflow. This study is based on our optical BVR, H alpha, and [O III] observations from the Wisconsin-Indiana-Yale-NRAO telescope and Owens Valley Radio Observatory CO interferometric data, along with available 5 GHz radio continuum and ROSAT X-ray maps.