A normal-mode theory for the dipole-exchange spin-wave spectrum in the finite-width ferromagnetic waveguide is presented. The theory takes into account a nonuniform character of the demagnetizing field in the waveguide cross section and, therefore, can be applied to any infinitely long, rectangular rod, even with square cross section. The inhomogeneity of static and dynamic dipole fields is taken into account using the same tensorial Green’s function, obtained from Maxwell equations, this fact allows to simplify the spectrum calculation procedure. According to the elaborated theory the spin-wave spectrum in the finite-width ferromagnetic waveguide can be calculated with simultaneous account of the dipole-dipole and exchange interaction, surface anisotropy, arbitrary direction of the external bias magnetic field and for any possible width-thickness aspect ratio of the magnonic waveguide. It is shown that the previously used analytical methods of the accounting of the finite width of the magnetic waveguides give unsuitable results for nanometer-size waveguides.