Geological and geochemical evidence indicates that the Antarctic ice sheet formed during the Eocene-Oligocene transition, 33.5-34.0 million years ago. Modelling studies suggest that such ice-sheet formation might have been triggered when atmospheric carbon dioxide levels (pCO2atm) fell below a critical threshold of approximately 750 p.p.m.v., but the timing and magnitude of pCO2atm relative to the evolution of the ice sheet has remained unclear. Here we use the boron isotope pH proxy on exceptionally well-preserved carbonate microfossils from a recently discovered geological section in Tanzania to estimate pCO2atm before, during and after the climate transition. Our data suggest that are reduction in pCO2atm occurred before the main phase of ice growth,followed by a sharp recovery to pre-transition values and then a more gradual decline. During maximum ice-sheet growth, pCO2atm was between approximately 450 and approximately 1,500 p.p.m.v., with a central estimate of approximately 760 p.p.m.v. The ice cap survived the period of pCO2atm recovery,although possibly with some reduction in its volume, implying (as models predict) a nonlinear response to climate forcing during melting. Overall, our results confirm the central role of declining pCO2atm in the development of the Antarctic ice sheet (in broad agreement with carbon cycle modelling) and help to constrain mechanisms and feedbacks associated with the Earth's biggest climate switch of the past 65 Myr.