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A mathematically rigorous and numerically efficient approach, based on analytical regularization, for solving the scalar wave diffraction problem with a Dirichlet boundary condition imposed on an arbitrarily shaped body of revolution is described. Seeking the solution in an integral-equation formulation, the singular features of its kernel are determined, and the initial equation transformed so that its kernel can be decomposed into a singular canonical part and a regular remainder. An analytical transformation technique is used to reduce the problem equivalently to an infinite system of linear algebraic equations of the second kind. Such system can be effectively solved with any prescribed accuracy by standard numerical methods. The matrix elements of this algebraic system are expressible in the terms of the Fourier coefficients of the remainder. Due to the smoothness of the remainder a robust and efficient technique is obtained to calculate the matrix elements. Numerical investigations of structures, such as the prolate spheroid and bodies obtained by rotation of "Pascal's Limaçon" and of the "Cassini Oval", exhibit the high accuracy and wide possibilities of the approach.