In this paper we deal with the von Neumann alternating projection method $x_{k+1}=P_{A}P_{B}x_{k}$ and with its generalization of the form $x_{k+1}=P_{A}(x_{k}+\lambda _{k}(P_{A}P_{B}x_{k}-x_{k}))$, where $A,B$ are closed and convex subsets of a Hilbert space $\mathcal{H}$ and $\operatorname{Fix}P_{A}P_{B}\neq \varnothing$. We do not suppose that $A\cap B\neq\varnothing$. We give sufficient conditions for the weak convergence of the sequence $(x_{k})$ to $\operatorname{Fix}P_{A}P_{B}$ in the general case and in the case $A$ is a closed affine subspace. We present also the results of preliminary numerical experiments.