We present a measurement of the sky distribution of positronium (Ps) annihilation continuum emission obtained with the SPI spectrometer on board ESAs INTEGRAL observatory. The only sky region from which significant Ps continuum emission is detected is the Galactic bulge. The Ps continuum emission is circularly symmetric about the Galactic centre, with an extension of about 8 deg FWHM. Within measurement uncertainties, the sky distribution of the Ps continuum emission is consistent with that found by us for the 511 keV electron-positron annihilation line using SPI. Assuming that 511 keV line and Ps continuum emission follow the same spatial distribution, we derive a Ps fraction of 0.92 +/- 0.09. These results strengthen our conclusions regarding the origin of positrons in our Galaxy based on observations of the 511 keV line. In particular, they suggest that the main source of Galactic positrons is associated with an old stellar population, such as Type Ia supernovae, classical novae, or low-mass X-ray binaries. Light dark matter is a possible alternative source of positrons.