Summary: Gill’s semigeostrophic equations [A. E. Gill, ibid. 80, 641–671 (1977; Zbl 0361.76102)] of one-layer, rotating channel hydraulics are exactly solved at the critical section, for flows with a finite uniform potential vorticity (PV). A detailed characterization of the critical flow at the sill is consequently obtained, together with a simple, explicit expression for the associated mass transport, that provides a direct extension of the well-known transport formula of zero-PV theory. The dependence of the mass transport on the PV, for a given average flow energy, is found to be non-monotonic. In particular, the transport increases when the PV is raised from zero, attaining a maximum value that can be up to twice those predicted by the zero-PV theory. This suggests that, contrary to common belief, finite-PV effects may be important in determining the mass transport in real oceanic outflows.