Some components of the Earth system may irreversibly collapse under global warming. Among them, the Atlantic Meridional Overturning Circulation (AMOC), the Greenland Ice Sheet, and West Antarctica Ice Sheet are of utmost importance for maintaining the present-day climate. In a simplified model, we show that both the rate of ice melting and the natural variability linked to freshwater fluxes over the Atlantic Ocean drastically affect how an ice sheet collapse impacts the AMOC stability.