The quantification of greenhouse gases present in the Archaean atmosphere is critical for understanding the evolution of atmospheric oxygen, surface temperatures and the conditions for life on early Earth. For instance, it has been argued that small changes in the balance between two potential greenhouse gases, carbon dioxide and methane, may have dictated the feedback cycle involving organic haze production and global cooling. Climate models have focused on carbon dioxide as the greenhouse gas responsible for maintaining above-freezing surface temperatures during a time of low solar luminosity. However, the analysis of 2.75-billion-year (Gyr)-old palaeosols--soil samples preserved in the geologic record--have recently provided an upper constraint on atmospheric carbon dioxide levels well below that required in most climate models to prevent the Earth's surface from freezing. This finding prompted many to look towards methane as an additional greenhouse gas to satisfy climate models. Here we use model equilibrium reactions for weathering rinds on 3.2-Gyr-old river gravels to show that the presence of iron-rich carbonate relative to common clay minerals requires a minimum partial pressure of carbon dioxide several times higher than present-day values. Unless actual carbon dioxide levels were considerably greater than this, climate models predict that additional greenhouse gases would still need to have a role in maintaining above-freezing surface temperatures.