||The coupled wavenumbers of a fluid-filled flexible cylindrical shell vibrating in the axisymmetric mode are studied. The coupled dispersion equation of the system is rewritten in the form of the uncoupled dispersion equation of the structure and the acoustic fluid, with an added fluid-loading term involving a parameter$\epsilon$ due to the coupling. Using the smallness of Poisson's ratio$(\nu )$, a double-asymptotic expansion involving$\epsilon$ and$\nu ^2$ is substituted in this equation. Analytical expressions are derived for the coupled wavenumbers (for large and small values of$\epsilon$). Different asymptotic expansions are used for different frequency ranges with continuous transitions occurring between them. The wavenumber solutions are continuously tracked as$\epsilon$ varies from small to large values. A general trend observed is that a given wavenumber branch transits from a rigid-walled solution to a pressure-release solution with increasing$\epsilon$. Also, it is found that at any frequency where two wavenumbers intersect in the uncoupled analysis, there is no more an intersection in the coupled case, but a gap is created at that frequency. Only the axisymmetric mode is considered. However, the method can be extended to the higher order modes.