The authors consider the parabolic integro-differential equation: \[ {\partial u\over \partial t} (x,t) + {\partial^2 u\over \partial x^2} (x,t) = \int^t_0 f(t,s,u(x,s))ds \] that is semidiscretized. The trapezoidal rule is adopted for the quadrature of the memory term. This equation arises in many applications such as compression of poroviscoelastic media, reactor dynamics, epidemic phenomena. The paper gives an error estimate for the fully discrete scheme with spectral method in space and the backward Euler method in time.