The snowball Earth theory, initially proposed by J.L. Kirschvink to explain the Neoproterozoic glacial episodes, suggests that the Earth was globally ice covered at 720 Ma (Sturtian episode) and 640 Ma (Marinoan episode). The reduction of the water cycle and the growth of large ice sheets led to a collapse of CO2 consumption through continental weathering and biological carbon pumping. As a consequence, atmospheric CO2 built up linearly to levels allowing escape from a snowball Earth. In this contribution, we question this assumed linear accumulation of CO2 into the atmosphere. Using a numerical model of the carbon-alkalinity cycles, we suggest that during global glaciations, even a limited area of open waters (103 km2) allows an efficient atmospheric CO2 diffusion into the ocean. This exchange implies that the CO2 consumption through the low-temperature alteration of the oceanic crust persists throughout the glaciation. Furthermore, our model shows that rising CO2 during the glaciation increases the efficiency of this sink through the seawater acidification. As a result, the atmospheric CO2 evolution is asymptotic, limiting the growth rate of the atmospheric carbon reservoir. Even after the maximum estimated duration of the glaciation (30 m.y.), the atmospheric CO2 is far from reaching the minimum deglaciation threshold (0.29 bar). Accounting for this previously neglected carbon sink, processes that decrease the CO2 deglaciation threshold must be further explored.