  Well-motivated models of dark matter often result in a population of
electrons and positrons within galaxies produced through dark matter
annihilation -- usually in association with gamma rays. As they diffuse through
galactic magnetic fields, these $e^\pm$ produce synchrotron radio emission. The
intensity and morphology of this signal depends on the properties of the
interstellar medium through which the $e^\pm$ propagate. Using observations of
the Andromeda Galaxy (M31) to construct a model of the gas, magnetic fields,
and starlight, we set constraints on dark matter annihilation to $b\bar{b}$
using the morphology of 3.6 cm radio emission. As the emission signal at the
center of M31 is very sensitive to the diffusion coefficient and dark matter
profile, we base our limits on the differential flux in the region between
$0.9-6.9$ kpc from the center. We exclude annihilation cross sections $\gtrsim
3 \times 10^{-25}$ cm$^3$/s in the mass range $10-500$ GeV, with a maximum
sensitivity of $7\times 10^{-26}$ cm$^3$/s at $20-40$ GeV. Though these limits
are weaker than those found in previous studies of M31, they are robust to
variations of the diffusion coefficient.
