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In this paper we investigate a discrete approximation in time and in space ofa Hilbert space valued stochastic process $\{u(t)\}_{t\in [0,T]}$ satisfying astochastic linear evolution equation with a positive-type memory term driven byan additive Gaussian noise. The equation can be written in an abstract form as$$ \dd u + (\int_0^t b(t-s) Au(s) \, \dd s)\, \dd t = \dd W^{_Q}, t\in (0,T];\quad u(0)=u_0 \in H, $$ where $W^{_Q}$ is a $Q$-Wiener process on$H=L^2({\mathcal D})$ and where the main example of $b$ we consider is given by$$ b(t) = t^{\beta-1}/\Gamma(\beta), \quad 0 < \beta <1. $$ We let $A$ be anunbounded linear self-adjoint positive operator on $H$ and we further assumethat there exist $\alpha >0$ such that $A^{-\alpha}$ has finite trace and that$Q$ is bounded from $H$ into $D(A^\kappa)$ for some real $\kappa$ with$\alpha-\frac{1}{\beta+1}<\kappa \leq \alpha$.  The discretization is achieved via an implicit Euler scheme and a Laplacetransform convolution quadrature in time (parameter $\Delta t =T/n$), and astandard continuous finite element method in space (parameter $h$). Let$u_{n,h}$ be the discrete solution at $T=n\Delta t$. We show that $$ (\E \|u_{n,h} - u(T)\|^2)^{1/2}={\mathcal O}(h^{\nu} + \Delta t^\gamma), $$ for any$\gamma< (1 - (\beta+1)(\alpha - \kappa))/2 $ and $\nu \leq\frac{1}{\beta+1}-\alpha+\kappa$.

Strong order of convergence of a fully discrete approximation of a linear stochastic Volterra type evolution equation