A single layer of fluid with thickness h(x) and geostrophic velocity u = (g'/f)∂(h + M)/∂x is conventionally used to model the flow of a density current over a sill/contraction, where M(x) is the cross-stream (x) bottom elevation, g' = gΔρ/ρ is reduced gravity, and f is the Coriolis parameter. The interface <h> intersects the parabolic bottom at two points x = ξ,L, with M(L) = H given. A new extremum for the volume discharge Q is obtained which is approximately half that of the Killworth-Whitehead upper bound, and which agrees with measurements in the Denmark Strait.