The aim of this paper is to study the numerical approximation of the weak solution to the stochastic Volterra type equation \[ dX(t)+\Big(\int_0^t b(t-s)AX(s)ds\Big)dt= dW^Q(t),\qquad t\in (0,T],\quad X(0)=X_0, \] on the real separable Hilbert space \(L^2(O)\), with \(O\) some convex polygonal domain in \(\mathbb R^d\). Here, \(W^Q\) is an \(H\)-valued Wiener process with covariance operator \(Q\). Letting \(V_h\subset H^1_0(O)\) consist of continuous piecewise linear functions vanishing at the boundary of \(O\), the authors apply the approximations \(X(t_n)\) defined via the equation \[ (V^n_h, v_h) - (X^{n-1}_h, v_h) +\Delta t\sum_{k=1}^n \omega _{n-k}(\nabla X^k_h, \nabla v_h)=(w^n, v_h) \] for \(n\geq 1\). The method used in the study of the weak convergence of the approximations relies on the Kolmogorov equation, but this method can not be applied directly in the case of the above equation because \(X(t)\), \(t\geq 0\), is not Markovian. The authors try to remove the drift and obtain an equation which has a Markovian solution. They also obtain a representation formula for the weak error and prove the main convergence result.