In this paper, we develop a new finite element model for a cable suspended in water. Global existence and uniqueness of solutions of the truncated system is shown for a slightly simplified equation describing the motion of a cable with negligible added mass and supported by fixed end-points. Based on this, along with well known results on local existence and uniqueness of solutions for symmetrizable hyperbolic systems, we conjecture a global result for the initial-boundary value problem. The FEM model for the cable is assembled to give a model of a multi-cable mooring system, which, in turn, is coupled to a rigid body model of the floating vessel. The result is a coupled dynamical model of a moored vessel, which can be applied to applications such as turret-based moored ships, or tension leg platforms.