A firm understanding of the relationship between atmospheric carbon dioxide concentration and temperature is critical for interpreting past climate change and for predicting future climate change. A recent synthesis suggests that the increase in global-mean surface temperature in response to a doubling of the atmospheric carbon dioxide concentration, termed `climate sensitivity', is between 1.5 and 6.2°C (5-95 per cent likelihood range), but some evidence is inconsistent with this range. Moreover, most estimates of climate sensitivity are based on records of climate change over the past few decades to thousands of years, when carbon dioxide concentrations and global temperatures were similar to or lower than today, so such calculations tend to underestimate the magnitude of large climate-change events and may not be applicable to climate change under warmer conditions in the future. Here we estimate long-term equilibrium climate sensitivity by modelling carbon dioxide concentrations over the past 420 million years and comparing our calculations with a proxy record. Our estimates are broadly consistent with estimates based on short-term climate records, and indicate that a weak radiative forcing by carbon dioxide is highly unlikely on multi-million-year timescales. We conclude that a climate sensitivity greater than 1.5°C has probably been a robust feature of the Earth's climate system over the past 420 million years, regardless of temporal scaling.