The northern continental slope of the Gulf of Mexico is riddled with numerous subsiding diapiric minibasins bounded by ridges, many but not all of which are connected by channels created by turbidity currents. The region is economically relevant in that many of these diapiric minibasins constitute focal points for the deposition of sand. Some of these sandy deposits in turn serve as excellent reservoirs for hydrocarbons. A better understanding of the "fill and spill" process by which minibasins fill with mud and sand as the intervening ridges are dissected by canyons may serve to aid in the location of such reservoirs. In the present paper a theory is developed to describe sediment deposition in minibasins. The theory relies on the hypotheses that the turbidity currents in question are sustained for at least about one hour. Two key and heretofore unrecognized aspects of the "fill and spill" process are revealed: (1) the formation of an internal hydraulic jump as a turbidity current spills into a confined basin, and (2) the detrainment of water across a settling interface forming at the top of the ponded turbidity current downstream of the hydraulic jump. It is shown that sufficiently strong detrainment can consume the flow, so that they is no outflow of either water or sediment even with continuous inflow. As the basin fills with sediment, however, overspill is eventually realized. The theory is developed into a numerical model, tested against experiments and applied at field scale in a companion paper.