  Threshold neutral pion photoproduction on the deuteron is studied in the
framework of baryon chiral perturbation theory beyond next-to--leading order in
the chiral expansion. To fourth order in small momenta, the amplitude is finite
and a sum of two- and three-body interactions with no undetermined parameters.
With accurate theoretical and experimental input from the single nucleon sector
for the proton amplitude, we investigate the sensitivity of the threshold cross
section to the elementary $\gamma n \to \pi^0 n$ amplitude. A precise
measurement of the threshold cross section for $\gamma d \to \pi^0 d$ is called
for.
