In this paper a complete mathematical solution of the problem of the buckling with large deflections of a thin circular plate under uniform radial pressure is given, assuming radial symmetry. Buckling takes place as soon as the prescribed thrust pe at the edge reaches a certain critical value pE. Thin plates do not, however, fail if the thrust pe is increased beyond pE. It is of importance, then, to determine the stresses when the ratio Λ = pe/pE becomes greater than unity. This problem, which is a nonlinear one, is solved by two methods for finite values of Λ; also an asymptotic solution is given for the limit state when Λ tends to infinity. The most notable single result is that the membrane stresses for large values of Λ become tensions in the interior of the plate and change abruptly to compressions in a narrow “boundary layer” at the edge of the plate. Curves showing the behavior of the deflection and the stresses are given for a series of values of Λ. Such curves show clearly, in particular, that the limit state is approached quite closely for relatively small values of Λ, i.e., Λ > 5. In closing, the authors discuss the relation between their results and von Kármán’s theory of effective width for the buckling of rectangular plates.