Analysis of sound transmission and penetration through a duct of a wall is one of the basic but important problems in structural acoustics. If the wall is thin, then the configuration may be regarded as a slit of zero thickness, and the treatment is rather easy. For the thick wall, however, one has to consider the duct propagation and the problem becomes somewhat involved. In this paper, plane-wave transmission through a two-dimensional duct of a rigid wall is treated rigorously and formulated as a boundary-value problem [K. Hongo and G. Ishii, IEEE Trans. Antennas Propag. AP-26, 494--499 (1978)]. First, the scattering field in each region is expressed in terms of appropriate eigenfunctions with unknown excitation coefficients. Then the continuity condition of the field at the aperture of the duct has been used for determining them. Starting from a rigorous solution, one can also derive a simple formula for estimating the transmission field through a small-aperture duct. Numerical computations have been done extensively and compared with those by other methods. Good agreements are found between this solution and others. Validity of the small-aperture approximation will also be discussed.