We analyze the fate of the Hesperian-aged outflow channel effluents emplaced into the northern lowlands of Mars. We have modeled the evolution of these effluents under the assumption that they were emplaced under a range of atmospheric conditions comparable to those of today and thought to have prevailed in the Hesperian. Under these conditions we find that the evolution of the sediment-loaded water after it left the channels includes three phases. Phase 1: Emplacement and intensive cooling: Violent emplacement of water followed by a short period of intensive evaporation from the surface and near-surface layer, and intensive convection. During this phase the water maintained and redistributed its large suspended sediment load. Water vapor strongly influenced the climate, at least for a geologically short time. When the temperature of the water reached the temperature of the maximum density or the freezing point, boiling and intensive convection ceased and the water deposited the sediments. Phase 2: Freezing solid: Geologically rapid freezing of the water body accompanied by weak convective water movement occurred over a period of the order of ~104 years. Phase 3: Sublimation and loss: This period involved sublimation of the ice and lasted longer than the freezing phase. The rate and latitudinal dependence of the sublimation, as well as the location of water vapor condensation, crucially depend on the planetary obliquity, climate, and sedimentary veneering of the ice. Phase 3 would have been very short geologically (~105--106 years) if an insulating sedimentary layer did not build up rapidly. If such an insulating layer did form rapidly, sublimation could cease and residual ice hundreds of meters thick could remain below the surface today. The northern lowlands of Mars are largely covered by the Late Hesperian-aged Vastitas Borealis Formation (VBF), which has sharp boundaries clearly seen in the map of kilometer-scale roughness, and a distinctive kilometer-scale roughness signature. We examine detrended topography data and find evidence that the VBF is underlain at very shallow depths by an Early Hesperian ridged volcanic plains substrate, rather than frozen water deposits remaining from the outflow channel events. Analysis of the VBF roughness characteristics suggests at least 100 m thickness of sediments on top of the underlying volcanic ridged plains. The total inferred volume of the VBF material approximately corresponds to the volume of material removed from the outflow channels. These results support a model in which the Vastitas Borealis Formation predominantly represents the sedimentary residue remaining after the sediment-laden water effluents of the outflow channels ponded in the northern lowlands and rapidly froze solid and sublimed. The distinctive facies of the VBF are interpreted to be related to the freezing, sublimation, and residue of the outflow channel effluents. Ridged facies are dominantly marginal and are interpreted to be related to ice sheet lateral retreat. Knobby facies occur throughout and are interpreted to be due to a variety of causes, including dewatering, de-icing and brine-related processes, and kame-like ice residues. Grooved facies are interpreted to represent postsublimation uplift and fracturing due to load removal. The mottled facies may represent bright ejecta whose albedo is related to buried sedimentary or evaporitic residues. Presently, the most likely sites to find frozen remnants of the Hesperian ocean are below the floors of stealth craters underlying the VBF. On the basis of these findings and interpretations, we make predictions for the fate of the outflow channel water vapor and the nature of the Martian hydrologic cycle during the Hesperian. If Noachian-aged oceans existed, conditions at that time would be similar to those described for a Hesperian ocean as soon as a global cryosphere was formed. If any residual deposits remain in the northern lowlands from a proposed Noachian ocean, they would lie below both the VBF and the underlying sequence of Hesperian ridged plains.