Using new and published data, we construct a sample of 160 brightest cluster galaxies (BCGs) spanning the redshift interval 0.03 < z < 1.63. We use this sample, which covers 70 per cent of the history of the universe, to measure the growth in the stellar mass of BCGs after correcting for the correlation between the stellar mass of the BCG and the mass of the cluster in which it lives. We find that the stellar mass of BCGs increases by a factor of 1.8 ± 0.3 between z = 0.9 and z = 0.2. Compared to earlier works, our result is closer to the predictions of semi-analytic models. However, BCGs at z = 0.9, relative to BCGs at z = 0.2, are still a factor of 1.5 more massive than the predictions of these models. Star formation rates in BCGs at z ∼ 1 are generally too low to result in significant amounts of mass. Instead, it is likely that most of the mass build up occurs through mainly dry mergers in which perhaps half of the mass is lost to the intra-cluster medium of the cluster.