Recent independent results from numerical simulations and observations have shown that brightest cluster galaxies (BCGs) have increased their stellar mass by a factor of almost 2 between z ∼ 0.9 and z ∼ 0.2. The numerical simulations further suggest that more than half this mass is accreted through major mergers. Using a sample of 18 distant galaxy clusters with over 600 spectroscopically confirmed cluster members between them, we search for observational evidence that major mergers do play a significant role. We find a major merger rate of 0.38 ± 0.14 mergers per Gyr at z ∼ 1. While the uncertainties, which stem from the small size of our sample, are relatively large, our rate is consistent with the results that are derived from numerical simulations. If we assume that this rate continues to the present day and that half of the mass of the companion is accreted on to the BCG during these mergers, then we find that this rate can explain the growth in the stellar mass of the BCGs that is observed and predicted by simulations. Major mergers therefore appear to be playing an important role, perhaps even the dominant one, in the build up of stellar mass in these extraordinary galaxies.