The stability of the Einstein universe is investigated using the new boundary conditions at surfaces of discontinuity, discovered by O'Brien and Synge. On certain assumptions it is shown that the Einstein model may represent a universe with inhomogeneities, but that, in general, these must be static otherwise the model is unstable. In the case of an instantaneous disturbance in the Einstein universe, the processes which cause instability must involve discontinuities in the pressure or its time derivatives. If the matter present satisfies an equation of state, no physically plausible disturbance of the static condition is possible, and the model is stable. If there is no equation of state, small disturbances involving changes in the pressure or its derivatives can occur, and will, in general, lead to instability, though in certain special cases condensations may begin to form without initial instability.