  We consider the phenomenology of the 511 keV emission in the Galactic bulge,
as recently observed by INTEGRAL, and propose a model is which the positrons
are produced by gamma-ray bursts (GRB) associated with mini starbursts in the
central molecular zone (CMZ). We show that the positrons can easily diffuse
across the bulge on timescales of about 10^7 yr, and that their injection rate
by GRBs is compatible with the observed fluxes if the mean time between two
GRBs in the bulge is about 8 10^4 yr x E_GRB_51. We also explain the low
disk-to-bulge emission ratio by noting that positrons from GRBs in the Galactic
disk should annihilate on timescales of < 10^4 yr in the dense shell of the
underlying supernova remnant, after the radiative transition, while the
remnants of GRBs occurring in the hot, low-density medium produced by recurrent
starbursts in the CMZ become subsonic before they can form a radiative shell,
allowing the positrons to escape and fill the whole Galactic bulge. If the mean
time between GRBs is smaller than 10^4 E_51 yr, INTEGRAL should be able to
detect the (localized) 511 keV emission associated with one or a few GRB
explosions in the disk.
