Using the “spring” model for a weak interface and the wave equations in cylindrical composite structures, we derive the generalized frequency equation for axisymmetric guided waves in a two-layered cylindrical composite structure with a weak interface- Numerical computations are made for dispersion characteristics of the axisymmetric guided waves in two-layered composite tubes(with or without water in hollow) and clad rods with rigid or slip interfaces- The influence of the shear stiffness of the weak interface on the lower order guided wave modes is studied-