Buoyancy-driven convection in a conditionally unstable ocean is studied theoretically. Conditionally unstable conditions are related to supercooled seawater. The freezing point is depressed due to increasing pressure, and upward motion (reduced pressure) leads to the formation of ice crystals in the form of frazil ice and hence a reduced bulk density of the rising mass parcel. For downward motion (increasing pressure), freezing does not occur. To model this one-way process, we take that rising parcels become lighter as they follow the adiabatic density lapse rate due to freezing. Through the action of a unit step function, we can model the fact that frazil ice is added only in the upward motion. Supercooled seawater is observed in the proximity of ice shelf fronts. We consider an idealized ice shelf and take the front to be vertical. For this geometry, we present linear analytical solutions as well as numerical results for nonlinear two-dimensional steady conditional convection in the presence of a stable environmental density gradient. With a parallel to moist convection in the atmosphere, we find convection cells near the ice front with rising fluid in a narrow region and sinking fluid over a much broader region.