  We prove convergence of a sequence of weak solutions of the nonlocal
Cahn-Hilliard equation to the strong solution of the corresponding local
Cahn-Hilliard equation. The analysis is done in the case of sufficiently smooth
bounded domains with Neumann boundary condition and a $W^{1,1}$-kernel. The
proof is based on the relative entropy method. Additionally, we prove the
strong $L^2$-convergence of the nonlocal operator to the negative Laplacian
together with a rate of convergence.
