The linear stability of an internal gravity wave of arbitrary amplitude in an unbounded stratified inviscid Boussinesq fluid is considered mathematically. The instability is shown to be governed by a Floquet system and treated by a generalization of the method of normal modes. Some properties of the Floquet system, and in particular those of its parametric instability, are analysed. The parametric instability is related to the theory of resonant wave interactions; and the surface of marginal stability in the control space of the amplitude and wavenumbers is shown to be describable by the catastrophe theory of Thom. Finally some results of numerical calculations of the marginal surface are shown. The main physical conclusion is to confirm that the internal gravity wave is unstable always, even when its amplitude is small and so its local Richardson number is large everywhere for all time. It is suggested, by various illustrations and arguments, that the methods developed in this paper are applicable to the instability of many symmetric nonlinear waves.