  Fermi measurements of the high-latitude gamma-ray background strongly
constrain a decaying-dark-matter origin for the 1--100 GeV Galactic positron
anomaly measured with PAMELA. Inverse-Compton scattering of the microwave
background by the emergent positrons produces a bump in the diffuse 100-200 MeV
gamma-ray background that would protrude from the observed background at these
energies. The positrons are thus constrained to emerge from the decay process
at a typical energy between ~100 GeV and ~250 GeV. By considering only
gamma-ray emission of the excess positrons and electrons, we derive a minimum
diffuse gamma-ray flux that, apart from the positron spectrum assumed, is
independent of the actual decay modes. Any gamma-rays produced directly by the
dark-matter decay leads to an additional signal that make the observational
limits more severe. A similar constraint on the energy of emergent positrons
from annihilation in dark-matter substructures is argued to exist, according to
recent estimates of enhancement in low-mass dark-matter substructures, and
improved simulations of such substructure will further sharpen this constraint.
