The Galactic positrons, as observed by their annihilation gamma-ray line at 0.511 MeV, are difficult to account for with astrophysical sources. It has been proposed that they are produced instead by dark matter annihilation or decay in the inner Galactic halo. To avoid other constraints, these processes are required to occur "invisibly," such that the eventual positron annihilation is the only detectable signal. However, electromagnetic radiative corrections to these processes inevitably produce real gamma rays ("internal bremsstrahlung"); this emission violates COMPTEL and EGRET constraints unless the dark matter mass is less than about 20 MeV.