The usual model for a sheath formed adjacent to a cathode immersed in a high pressure plasma is modified for ionization and secondary emission effects. It is found at high voltages, when the product of the ionization coefficient within the sheath and the sheath thickness is of order unity, that the sheath can collapse catastrophically with an accompanying decrease in its impedance of many orders of magnitude. The calculations, which are numerical, are particularly well suited to the usual description of the ionization coefficient in terms of two empirical constants, appropriate to a given gas. As a result, it is possible to give a simple curve of sheath breakdown voltage versus the quantity ji0/μiP3 (where ji0 is the ion current density, μi the ion mobility and p the pressure). The breakdown curve for air is given as an example.