The Milky Way Galaxy glows with the soft gamma ray emission resulting from the annihilation of ~5 × 1043 electron-positron pairs every second. The origin of this vast quantity of antimatter and the peculiar morphology of the 511keV gamma ray line resulting from this annihilation have been the subject of debate for almost half a century. Most obvious positron sources are associated with star forming regions and cannot explain the rate of positron annihilation in the Galactic bulge, which last saw star formation some 10 Gyr ago, or else violate stringent constraints on the positron injection energy. Radioactive decay of elements formed in core collapse supernovae (CCSNe) and normal Type Ia supernovae (SNe Ia) could supply positrons matching the injection energy constraints but the distribution of such potential sources does not replicate the required morphology. We show that a single class of peculiar thermonuclear supernova - SN1991bg-like supernovae (SNe 91bg) - can supply the number and distribution of positrons we see annihilating in the Galaxy through the decay of 44Ti synthesised in these events. Such 44Ti production simultaneously addresses the observed abundance of 44Ca, the 44Ti decay product, in solar system material.