The Last Glacial Maximum (LGM), one of the best studied palaeoclimatic intervals, offers an excellent opportunity to investigate how the climate system responds to changes in greenhouse gases and the cryosphere. Previous work has sought to constrain the magnitude and pattern of glacial cooling from palaeothermometers1,2, but the uneven distribution of the proxies, as well as their uncertainties, has challenged the construction of a full-field view of the LGM climate state. Here we combine a large collection of geochemical proxies for sea surface temperature with an isotope-enabled climate model ensemble to produce a field reconstruction of LGM temperatures using data assimilation. The reconstruction is validated with withheld proxies as well as independent ice core and speleothem δ18O measurements. Our assimilated product provides a constraint on global mean LGM cooling of -6.1 degrees Celsius (95 per cent confidence interval: -6.5 to -5.7 degrees Celsius). Given assumptions concerning the radiative forcing of greenhouse gases, ice sheets and mineral dust aerosols, this cooling translates to an equilibrium climate sensitivity of 3.4 degrees Celsius (2.4-4.5 degrees Celsius), a value that is higher than previous LGM-based estimates but consistent with the traditional consensus range of 2-4.5 degrees Celsius3,4.