The linear stability of an axially symmetric model of a warm-core eddy over a stratified ocean of infinite depth is investigated using asymptotic techniques for large wavenumber and small Burger number. The development is similar to previous work on the stability of a geostrophic front (Kroll, 1992) and the results found there can be modified and applied to most oceanic eddies. The most important result is that instability occurs in a region confined at the edge of the eddy with maximum width of a fraction of the radius given approximately by the Rossby number for realistic eddies. We expect the instability to produce turbulence and contribute to the breakdown of the interface between the eddy and the surrounding ocean in this region.