A boundary integral method for solving the exterior acoustic radiation problem of axisymmetric bodies with arbitrary boundary conditions is developed. The new formulation derives from the method proposed by Burton and Miller [Proc. R. Soc. London Ser. A323, 201–210 (1971)], which uses a linear combination of the Helmholtz integral equation and its normal derivative to ensure the uniqueness of the numerical solution at all frequencies. It is shown that the use of tangential operators enables one to easily compute the hypersingular integral without much extra effort. By taking advantage of the properties of axisymmetric geometry, and using the expansion of the boundary conditions and the surface distribution functions in Fourier series with respect to the angle of revolution, the surface integral is reduced to a line integral along the generator of the body, and Fourier integrals of the Green’s function and its derivatives over the circumferential angle. The Fourier integrals with singular kernels are evaluated using recurrence formulas in terms of the complete elliptic integrals. A new approach is then proposed to evaluate the Burton and Miller formulation at a corner by using the left-hand and right-hand limits of normal derivatives. Doing so, isoparametric elements throughout the surface of the body are able to be used. In order to demonstrate the validity and accuracy of the method, numerical results with quadratic isoparametric curvilinear elements are presented for radiation problems of a pulsating sphere, an oscillating sphere, and a finite cylinder with various boundary conditions.