Three-dimensional underwater vibrations of a geometrically non-linear cable with a weight at the lower end are investigated. The length of the cable is time-dependent. A set of non-linear, time-varying differential equations describing this system is derived by Hamilton's principle and the variable-domain finite element method. The results of numerical simulation are presented for constant-speed and sinusoidal axial motions of the cable. The vibration responses of three initial conditions are shown. The effects of initial displacements, initial tensions due to gravity and the hydrodynamic forces on the non-linear longitudinal and transverse amplitudes are presented. Finally, some conclusions are drawn.