Low-density condensates are often modelled by the Gross-Pitaevskii equation, which is a nonlinear Schrödinger equation determining the one-particle wave function,ψ. Its solution can be interpreted hydrodynamically in terms of a compressible fluid of density ρ, given by the squared amplitude of psi;, moving irrotationally with a velocity that is the gradient of a potential ϕ given by the argument of psi;. Sinceψ is single valued, ϕ is arbitrary to an arbitrary additive multiple of 2л. The circulation round a vortex line, i.e. round a curve on which psi; =<ρ =0, is, therefore, quantized in units ofκ =h/M, where M is the particle mass and h is Planck's constant. This paper analyses waves of infinitesimal amplitude travelling on a singly quantized rectilinear vortex. It is therefore a sequel to the previous paper in this series in which Kelvin waves on a hollow-core vortex in a compressible Euler fluid were investigated. In both situations, the waves may be bound or free, depending on their angular and axial wavenumbers, m and k. The free waves radiate energy acoustically to infinity, while the bound states do not. It is found that one class of m =1 modes, called‘slow modes’because of their low frequency, consists of bound states for all k. The slow m =2 modes are also bound when k is sufficiently large, but are free for small k.