  A del Pezzo surface of degree one defined over the rationals has 240
exceptional curves. These curves are permuted by the action of the absolute
Galois group. We show how a solution to the classical inverse Galois problem
for a subgroup of the Weyl group of type $D_8$ gives rise to a solution of the
inverse Galois problem for the action of this subgroup on the 240 exceptional
curves. A del Pezzo surface of degree one with such a Galois action contains a
Galois invariant sublattice of type $D_8$ within its Picard lattice; this can
be characterized in terms of a certain set of sixteen points in $\mathbb{P}^4$.
