  The numerical approximation of the solution to a stochastic partial
differential equation with additive spatial white noise on a bounded domain is
considered. The differential operator is assumed to be a fractional power of an
integer order elliptic differential operator. The solution is approximated by
means of a finite element discretization in space and a quadrature
approximation of an integral representation of the fractional inverse from the
Dunford-Taylor calculus.
  For the resulting approximation, a concise analysis of the weak error is
performed. Specifically, for the class of twice continuously Fr\'echet
differentiable functionals with second derivatives of polynomial growth, an
explicit rate of weak convergence is derived, and it is shown that the
component of the convergence rate stemming from the stochasticity is doubled
compared to the corresponding strong rate. Numerical experiments for different
functionals validate the theoretical results.
