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The sinking and recirculation of Antarctic Bottom Water (AABW) are a major regulator of the storage of heat, carbon, and nutrients in the ocean. This sinking is sensitive to changes in surface buoyancy, in particular because of freshening since salinity plays a greater role in determining density at cold temperatures. Acceleration in Antarctic ice-shelf and land-ice melt could thus significantly impact the ventilation of the world's oceans, yet future projections do not usually include this effect in models. Here we use an ocean-sea ice model to investigate the potential long-term impact of Antarctic meltwater on ocean circulation and heat storage. The freshwater forcing is derived from present-day estimates of meltwater input from drifting icebergs and basal melt, combined with RCP2.6, RCP4.5, and RCP8.5 scenarios of projected amplification of Antarctic meltwater. We find that the additional freshwater induces a substantial slowdown in the formation rate of AABW, reducing ventilation of the abyssal ocean. Under both the RCP4.5 and RCP8.5 meltwater scenarios, there is a near-complete shutdown of AABW formation within just 50 years, something that is not captured by climate model projections. The abyssal overturning at similar to 30 degrees S also weakens, with an similar to 20-yr delay relative to the onset of AABW slowdown. After 200 years, up to similar to 50% of the original volume of AABW has disappeared as a result of abyssal warming, induced by vertical mixing in the absence of AABW ventilation. This result suggests that climate change could induce the disappearance of present-day abyssal water masses, with implications for the global distribution of heat, carbon, and nutrients.