The Fermi surface-the set of points in momentum space describing gapless electronic excitations-is a central concept in the theory of metals. In this context, the normal ＇metallic＇ state of the optimally doped high-temperature superconductors is not very unusual : above the superconducting transition temperature, T-c, there is evidence for a large Fermi surface(1-3) despite the absence of well-defined elementary excitations. In contrast, the normal state of underdoped high-temperature superconductors differs in that there is evidence for a ＇pseudogap＇ above T-c (refs 4-6). Here we examine, using angle-resolved photoemission spectroscopy, the temperature dependence of the Fermi surface in underdoped Bi2Sr2CaCu2O8+delta. We find that, on cooling the sample, the pseudogap opens up at different temperatures for different points in momentum space. This leads to an initial breakup of the Fermi surface, at a temperature T*, into disconnected arcs, which then shrink with decreasing temperature before collapsing to the point nodes of the superconducting ground state below T-c. This unusual behaviour, where the Fermi surface does not form a continuous contour in momentum space as in conventional metals, is unprecedented in that it occurs in the absence of long-range order. Moreover, although the superconducting gap below T-c evolves smoothly into the pseudogap above T-c, the pseudogap differs in its unusual temperature-dependent anisotropy, implying an intimate but non-trivial relationship between the pseudogap and the superconducting gap.