  The solution to the initial and Dirichlet boundary value problem for a
semilinear, one dimensional heat equation is approximated by a numerical method
that combines the Besse relaxation scheme in time (C. R. Acad. Sci. Paris
S{\'e}r. I, vol. 326 (1998)) with a central finite difference method in space.
A new, composite stability argument is developed, leading to an optimal,
second-order error estimate in the discrete $L_t^{\infty}(H_x^1)-$norm. It is
the first time in the literature where an error estimate for fully discrete
approximations based on the Besse relaxation scheme is provided.
