  The intense 0.511 MeV gamma-ray line emission from the Galactic Center
observed by INTEGRAL requires a large annihilation rate of nonrelativistic
positrons. If these positrons are injected at even mildly relativistic
energies, higher-energy gamma rays will also be produced. We calculate the
gamma-ray spectrum due to inflight annihilation and compare to the observed
diffuse Galactic gamma-ray data. Even in a simplified but conservative
treatment, we find that the positron injection energies must be $\lesssim 3$
MeV, which strongly constrains models for Galactic positron production.
