  We present a scheme for laser cooling applicable for an extremely dilute
sample of magnetically trapped antihydrogen atoms($\bar{H}$). Exploiting and
controlling the dynamical coupling between the $\bar{H}$'s motional degrees of
freedom in a magnetic trap, three-dimensional cooling can be achieved from
Doppler cooling on one dimension using the $1s_{1/2}-2p_{3/2}$ transition. The
lack of three-dimensional access to the trapped $\bar{H}$ and the nearly
separable nature of the trapping potential leads to difficulties in cooling.
Using realistic models for the spatial variation of the magnetic fields, we
find that it should be possible to cool the $\bar{H}$'s to $\sim 20$ mK even
with these constraints.
