  Our Galaxy hosts the annihilation of a few $\times 10^{43}$ low-energy
positrons every second. Radioactive isotopes capable of supplying such
positrons are synthesised in stars, stellar remnants, and supernovae. For
decades, however, there has been no positive identification of a main stellar
positron source leading to suggestions that many positrons originate from
exotic sources like the Galaxy's central super-massive black hole or dark
matter annihilation. %, but such sources would not explain the
recently-detected positron signal from the extended Galactic disk. Here we show
that a single type of transient source, deriving from stellar populations of
age 3-6 Gyr and yielding ~0.03 $M_\odot$ of the positron emitter $^{44}$Ti, can
simultaneously explain the strength and morphology of the Galactic positron
annihilation signal and the solar system abundance of the $^{44}$Ti decay
product $^{44}$Ca. This transient is likely the merger of two low-mass white
dwarfs, observed in external galaxies as the sub-luminous, thermonuclear
supernova known as SN1991bg-like.
