In this paper we investigate a discrete approximation in time and in space of a Hilbert space valued stochastic process {u(t)}t∈[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 du+ (∫ t 0 b(t− s)Au(s) ds ) dt = dW , t ∈ (0, T ]; u(0) = u0 ∈ H, where W is a Q-Wiener process on H = L(D) and where the main example of b we consider is given by b(t) = tβ−1/Γ (β), 0 0 such that A−α has finite trace and that Q is bounded from H into D(A) for some real κ with α− 1 β+1 < κ ≤ α. The discretization is achieved via an implicit Euler scheme and a Laplace transform convolution quadrature in time (parameter ∆t = T/n), and a standard continuous finite element method in space (parameter h). Let un,h be the discrete solution at T = n∆t. We show that ( E‖un,h − u(T )‖ )1/2 = O(h +∆t), for any γ < (1− (β + 1)(α− κ))/2 and ν ≤ 1 β+1 − α+ κ. M. Kovacs Department of Mathematics and Statistics, University of Otago, PO Box 56, Dunedin, 9054, New Zealand. E-mail: mkovacs@maths.otago.ac.nz J. Printems Laboratoire d’Analyse et de Mathematiques Appliquees CNRS UMR 8050, 61, avenue du General de Gaulle, Universite Paris–Est, 94010 Creteil, France. E-mail: printems@u-pec.fr 2 Mihaly Kovacs, Jacques Printems