Maps of the Galactic electron-positron annihilation radiation show evidence for three distinct and significant features: (1) a central bulge source, (2) emission in the Galactic plane, and (3) an enhancement of emission at positive latitudes above the Galactic Center. In this paper, we explore the possibility that Sgr A East, a very prominent radio structure surrounding the Galactic nucleus, may be a significant contributer to the central bulge feature. The motivation for doing so stems from a recently proposed link between this radio object and the EGRET gamma -ray source 2EG J1746-2852. If this association is correct, then Sgr A East is also expected to be a source of copious positron production. The results presented here show that indeed Sgr A East must have produced a numerically significant population of positrons, but also that most of them have not yet had sufficient time to thermalize and annihilate. As such, Sgr A East by itself does not appear to be the dominant current source of annihilation radiation, but it will be when the positrons have cooled sufficiently and they have become thermalized. This raises the interesting possibility that the bulge component may be due to the relics of earlier explosive events like the one that produced Sgr A East.