Bernal (AB)-stacked bilayer graphene (BLG) is a semiconductor whose bandgap can be tuned by a transverse electric field, making it a unique material for a number of electronic and photonic devices(1-3). A scalable approach to synthesize high-quality BLG is therefore critical, which requires minimal crystalline defects in both graphene layers(4,5) and maximal area of Bernal stacking, which is necessary for bandgap tunability(6). Here we demonstrate that in an oxygen-activated chemical vapour deposition (CVD) process, half-millimetre size, Bernal-stacked BLG single crystals can be synthesized on Cu. Besides the traditional 'surface-limited' growth mechanism for SLG (1st layer), we discovered new microscopic steps governing the growth of the 2nd graphene layer below the 1st layer as the diffusion of carbon atoms through the Cu bulk after complete dehydrogenation of hydrocarbon molecules on the Cu surface, which does not occur in the absence of oxygen. Moreover, we found that the efficient diffusion of the carbon atoms present at the interface between Cu and the 1st graphene layer further facilitates growth of large domains of the 2nd layer. The CVD BLG has superior electrical quality, with a device on/off ratio greater than 10(4), and a tunable bandgap up to similar to 100 meV at a displacement field of 0.9 V nm(-1).