  In this paper we investigate a discrete approximation in time and in space of
a Hilbert space valued stochastic process $\{u(t)\}_{t\in [0,T]}$ satisfying a
stochastic linear evolution equation with a positive-type memory term driven by
an 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 an
unbounded linear self-adjoint positive operator on $H$ and we further assume
that 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 Laplace
transform convolution quadrature in time (parameter $\Delta t =T/n$), and a
standard 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 $\nu \leq
\frac{1}{\beta+1}-\alpha+\kappa$.
