Two different model-independent mapping techniques have been applied to Compton Gamma Ray Observatory OSSE, SMM, TGRS, and balloon data and reveal a feature in the 0.511 MeV e(+)-e(-) annihilation radiation pattern of our Galaxy centered at I similar to -2 degrees and b similar to 10 degrees with a flux of approximately 5 x 10(-4) 0.511 MeV photons cm(-2) s(-1) If near the Galactic center, then positron (e(+)) sources are producing about 10(42) e(+) s(-1), which annihilate approximate to 1-2 kpc above the Galactic plane. A starburst episode within the inner few hundred parsecs of our Galaxy would drive hot pair-laden gas into the halo, with the one-sidedness pointing to the site of initial pressure release at the onset of the starburst activity. Positrons lose energy and annihilate as they are convected upward with the gas flow, and we calculate high-latitude annihilation patterns and fluxes in accord with the observations. Changes in the ionization state when the escaping gas cools could give annihilation radiation substructure. The fountain of hot (similar to 10(6)-10(7) K) gas rising into the Galactic halo would be seen through its enhanced dispersion measure, thermal emission, and recombination radiation.