Gamma-ray line radiation at 511keV is the signature of electron-positron annihilation. Such radiation has been known for 30years to come from the general direction of the Galactic Centre, but the origin of the positrons has remained a mystery. Stellar nucleosynthesis, accreting compact objects, and even the annihilation of exotic dark-matter particles have all been suggested. Here we report a distinct asymmetry in the 511-keV line emission coming from the inner Galactic disk (~10-50° from the Galactic Centre). This asymmetry resembles an asymmetry in the distribution of low mass X-ray binaries with strong emission at photon energies >20keV (`hard' LMXBs), indicating that they may be the dominant origin of the positrons. Although it had long been suspected that electron-positron pair plasmas may exist in X-ray binaries, it was not evident that many of the positrons could escape to lose energy and ultimately annihilate with electrons in the interstellar medium and thus lead to the emission of a narrow 511-keV line. For these models, our result implies that up to a few times 1041 positrons escape per second from a typical hard LMXB. Positron production at this level from hard LMXBs in the Galactic bulge would reduce (and possibly eliminate) the need for more exotic explanations, such as those involving dark matter.