Current global warming necessitates a detailed understanding of the relationships between climate and global ice volume. Highly resolved and continuous sea-level records are essential for quantifying ice-volume changes. However, an unbiased study of the timing of past ice-volume changes, relative to polar climate change, has so far been impossible because available sea-level records either were dated by using orbital tuning or ice-core timescales, or were discontinuous in time. Here we present an independent dating of a continuous, high-resolution sea-level record(1,2) in millennial-scale detail throughout the past 150,000 years. We find that the timing of ice-volume fluctuations agrees well with that of variations in Antarctic climate and especially Greenland climate. Amplitudes of ice-volume fluctuations more closely match Antarctic (rather than Greenland) climate changes. Polar climate and ice-volume changes, and their rates of change, are found to covary within centennial response times. Finally, rates of sea-level rise reached at least 1.2 m per century during all major episodes of ice-volume reduction.