The 24 MJup brown dwarf 2MASS 1207-3932 has for some time been known to show clear signs of classical T Tauri-like accretion. Through analysis of its oxygen forbidden emission, we have discovered that it is driving a bipolar outflow. Blue- and redshifted components to the [O I] λ6300 forbidden emission line are seen at velocities of -8 and +4 km s-1 (on either side of the systemic velocity). Spectroastrometry recovers the position of both components relative to the brown dwarf at ~0.08'' (in opposing directions). A position-velocity diagram of the line region supports the spectroastrometric results. The Hα and He I λ6678 lines were also analyzed. These line regions are not offset with respect to the continuum, ruling out the presence of spectroastrometric artifacts and underlining the validity of the [O I] λ6300 results. The low radial velocity of the outflow and the relatively large offsets are consistent with 2MASS 1207-3932 having a near edge-on disk as proposed by Scholz et al. 2MASS 1207-3932 is now the smallest mass galactic object known to drive an outflow. The age of the TW Hydrae association (~8 Myr) also makes this one of the oldest objects with a resolved jet. This discovery not only highlights the robustness of the outflow mechanism over an enormous range of masses but also suggests that it may even be feasible for young giant planets with accretion disks to drive outflows.