The ocean depth at which the rate of calcium carbonate input from surface waters equals the rate of dissolution is termed the calcite compensation depth. At present, this depth is approximately 4,500 m, with some variation between and within ocean basins. The calcite compensation depth is linked to ocean acidity, which is in turn linked to atmospheric carbon dioxide concentrations and hence global climate. Geological records of changes in the calcite compensation depth show a prominent deepening of more than 1 km near the Eocene/Oligocene boundary (approximately 34 million years ago) when significant permanent ice sheets first appeared on Antarctica, but the relationship between these two events is poorly understood. Here we present ocean sediment records of calcium carbonate content as well as carbon and oxygen isotopic compositions from the tropical Pacific Ocean that cover the Eocene/Oligocene boundary. We find that the deepening of the calcite compensation depth was more rapid than previously documented and occurred in two jumps of about 40,000 years each, synchronous with the stepwise onset of Antarctic ice-sheet growth. The glaciation was initiated, after climatic preconditioning, by an interval when the Earth's orbit of the Sun favoured cool summers. The changes in oxygen-isotope composition across the Eocene/Oligocene boundary are too large to be explained by Antarctic ice-sheet growth alone and must therefore also indicate contemporaneous global cooling and/or Northern Hemisphere glaciation.