The Hall coefficient RH of the cuprate superconductor YBa 2Cu3Oy was measured in magnetic fields up to 60 T for a hole concentration p from 0.078 to 0.152 in the underdoped regime. In fields large enough to suppress superconductivity, RH(T) is seen to go from positive at high temperature to negative at low temperature, for p0.08. This change of sign is attributed to the emergence of an electron pocket in the Fermi surface at low temperature. At p<0.08, the normal-state R H(T) remains positive at all temperatures, increasing monotonically as T→0. We attribute the change of behavior across p=0.08 to a Lifshitz transition, namely a change in Fermi-surface topology occurring at a critical concentration pL=0.08, where the electron pocket vanishes. The loss of the high-mobility electron pocket across pL coincides with a tenfold drop in the conductivity at low temperature, revealed in measurements of the electrical resistivity ρ at high fields, showing that the so-called metal-insulator crossover of cuprates is in fact driven by a Lifshitz transition. It also coincides with a jump in the in-plane anisotropy of ρ, showing that without its electron pocket, the Fermi surface must have strong twofold in-plane anisotropy. These findings are consistent with a Fermi-surface reconstruction caused by a unidirectional spin-density wave or stripe order. © 2011 American Physical Society.