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, Tc, 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 sub c [4-6]. Here we examine, using angle-resolved photoemission spectroscopy, the temperature dependence of the Fermi surface in underdoped Bi2 Sr2 CaCu2 O8+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 Tc. 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 Tc evolves smoothly into the pseudogap above Tc, the pseudogap differs in its unusual temperature-dependent anisotropy, implying an intimate but non-trivial relationship between the pseudogap and the superconducting gap.