In this paper, we study the mathematical formulation for an optimal
control problem governed by a linear parabolic integro-differential
equation and present the optimality conditions. We then set up its
weak formulation and the finite element approximation scheme. Based
on these we derive the a priori error estimates for its finite
element approximation both in $H^1$ and $L^2$ norms. Furthermore
some numerical tests are presented to
verify the theoretical results.