The author considers the initial-boundary-value problem for the nonlinear parabolic integro-differential equation \[ qu_t= \nabla\cdot\Biggl(a\Delta u+ \int^t_0b\Delta u(\tau) d\tau\Biggr)= f,\quad (t,x)\in [0,T]\times \Omega, \] where \(q= (q(t,x,u)\), \(a= a(u)\), \(b= b(t,u(\tau),\nabla u(\tau))\), \(f= f(u,\nabla u)\), and \(\Omega= I_1\times I_2\times I_3\in \mathbb{R}^3\). It is solved numerically the alternating direction implicit (ADI) Galerkin scheme, employing local approximations based on patches of finite elements. The derivation of optimal \(H^1\)- and \(L^2\)-error estimates makes use of the Ritz-Volterra projection [see Y. Lin, V. Thomée and L. B. Wahlbin, SIAM J. Numer. Anal. 28, No. 4, 1047-1070 (1991; Zbl 0728.65117)] for treating the integral term. The paper concludes with comments on how to initialize the scheme and on extensions of the results to more general problems including spatial dependency in the given data \(a\), \(b\) and \(f\).