  In this paper we study the approximation of the distribution of $X_t$
Hilbert--valued stochastic process solution of a linear parabolic stochastic
partial differential equation written in an abstract form as $$ dX_t+AX_t dt =
Q^{1/2} d W_t, \quad X_0=x \in H, \quad t\in[0,T], $$ driven by a Gaussian
space time noise whose covariance operator $Q$ is given. We assume that
$A^{-\alpha}$ is a finite trace operator for some $\alpha>0$ and that $Q$ is
bounded from $H$ into $D(A^\beta)$ for some $\beta\geq 0$. It is not required
to be nuclear or to commute with $A$. The discretization is achieved thanks to
finite element methods in space (parameter $h>0$) and implicit Euler schemes in
time (parameter $\Delta t=T/N$). We define a discrete solution $X^n_h$ and for
suitable functions $\phi$ defined on $H$, we show that $$ |\E \phi(X^N_h) - \E
\phi(X_T) | = O(h^{2\gamma} + \Delta t^\gamma) $$ \noindent where $\gamma<1-
\alpha + \beta$. Let us note that as in the finite dimensional case the rate of
convergence is twice the one for pathwise approximations.
