  Gamma ray emission of 511 keV lines arising from electron-positron
annihilation has been detected from the Galaxy since the 70s. Spatially
resolved observations using the INTEGRAL satellite have shown its full sky
distribution to be strongly concentrated in the Galactic bulge, with a smaller
contribution from the disk, unlike the situation in any other wavelength. The
puzzling distribution of the positrons gave rise to various suggestions,
including stellar nucleosynthesis in core-collapse (CC) and type Ia
thermonuclear supernovae (SNe), accreting compact objects, and more "exotic"
explanations of annihilation of dark-matter particles. However, such models
encounter difficulties in reproducing the total Galactic 511 keV flux as well
as its peculiar bulge-centered distribution. Theoretical models of SNe from
thermonuclear Helium detonations on white dwarfs (WDs) were suggested as
potential additional sources of positrons, contributing to the Galactic
Gamma-ray emission. Here, we show that the recently discovered class of faint,
calcium-rich type Ib SNe (with the prototype SN 2005E), thought to arise from
such explosions, also called ".Ia" SNe, can explain both the 511 keV flux and
its distribution, and can eliminate the need for non-astrophysical (e.g., dark
matter annihilation) sources. Such SNe comprise a fraction of only ~2% of all
SNe, but they currently inject hundreds of times more positrons (from $^{44}$Ti
decay) to the ISM than injected by CC SNe, enough to reproduce the total
Galactic 511 keV line emission. They exclusively explode in old environments,
and their contribution to the 511 keV emission therefore follows the old (>10
Gyrs) Galactic stellar population, dominated by the bulge, thereby reproducing
the observed large bulge-to-disk ratio.
