The stability analysis of the International Thermonuclear Experimental Reactor (ITER) Central Solenoid (CS) coil Nb3Sn conductor is performed following a similar approach to that recently used for the ITER Toroidal Field and Poloidal Field conductors. The most critical conductors in the winding pack, as well as the most critical (minimum temperature margin) location along them, are identified by the application of the Vincenta code, which also provides the initial and boundary conditions in the reference case. The Mithrandir code is then applied to these conductors, using a much finer grid than affordable in the Vincenta analysis, in order to capture the details of normal zone initiation and possible recovery to SC state when different square wave disturbances of length L in the range [0.01 m, 7 m] and duration τ in the range [1 ms, 100 ms] are applied to the superconducting cable. The computed minimum quench energy is shown to be typically above (in one case borderline to) the expected disturbance. The sensitivity to parametric variations of the heat transfer coefficient between strands and helium is significant for some disturbances. The inclusion of an external circuit in the model, providing selfconsistent boundary conditions, does not influence the results.