One nonlocal boundary value problem for the Poisson equation in an rectangular domain is considered in this work. The concept of strong solution of this problem is introduced. The correct solvability of this problem in weighted Sobolev spaces generated by a mixed norm is proved using the Fourier method. This is accomplished by exploiting the basis property of the system of eigenfunctions and associated functions of the corresponding spectral problem in weighted Lebesgue spaces, as well as the Hausdorff-Young inequality valid for this system.