A hard cylindrical waveguide, filled with a uniform acoustic medium, has a circular cross section. A hard sphere is located with its centre on the axis of the cylinder. We look for a trapped mode, i.e. a mode of finite energy such that the normal velocity vanishes on both the sphere and the cylinder. The velocity potentials considered here have an angular variation exp (i$\alpha $) about the axis of symmetry, where $\alpha $ is the azimuthal angle; it is well known that modes of this type cannot propagate to infinity if their wavenumber lies below a critical cut-off wavenumber. It is assumed that the radius of the sphere is sufficiently small and that the wavenumber lies just below the cut-off wavenumber; it is shown by an explicit construction that a trapped mode exists when there is a certain characteristic relation between the radius of the sphere and the wavenumber of the mode.