Tropospheric ozone (O-3) is a key component of air pollution and an important anthropogenic greenhouse gas(1). During the twentieth century, the proliferation of the internal combustion engine, rapid industrialization and land-use change led to a global-scale increase in O-3 concentrations(2,3); however, the magnitude of this increase is uncertain. Atmospheric chemistry models typically predict(4-7) an increase in the tropospheric O-3 burden of between 25 and 50 per cent since 1900, whereas direct measurements made in the late nineteenth century indicate that surface O-3 mixing ratios increased by up to 300 per cent(8-10) over that time period. However, the accuracy and diagnostic power of these measurements remains controversial(2). Here we use a record of the clumped-isotope composition of molecular oxygen ((OO)-O-18-O-18 in O-2) trapped in polar firn and ice from 1590 to 2016 ad, as well as atmospheric chemistry model simulations, to constrain changes in tropospheric O-3 concentrations. We find that during the second half of the twentieth century, the proportion of (OO)-O-18-O-18 in O-2 decreased by 0.03 +/- 0.02 parts per thousand (95 per cent confidence interval) below its 1590-1958 ad mean, which implies that tropospheric O-3 increased by less than 40 per cent during that time. These results corroborate model predictions of global-scale increases in surface pollution and vegetative stress caused by increasing anthropogenic emissions of O-3 precursors(4,5,11). We also estimate that the radiative forcing of tropospheric O-3 since 1850 ad is probably less than +0.4 watts per square metre, consistent with results from recent climate modelling studies(12).