  We prove a weak error estimate for the approximation in space and time of a
semilinear stochastic Volterra integro-differential equation driven by additive
space-time Gaussian noise. We treat this equation in an abstract framework, in
which parabolic stochastic partial differential equations are also included as
a special case. The approximation in space is performed by a standard finite
element method and in time by an implicit Euler method combined with a
convolution quadrature. The weak rate of convergence is proved to be twice the
strong rate, as expected. Our convergence result concerns not only functionals
of the solution at a fixed time but also more complicated functionals of the
entire path and includes convergence of covariances and higher order
statistics. The proof does not rely on a Kolmogorov equation. Instead it is
based on a duality argument from Malliavin calculus.
