  Despite twenty years of research, the phase diagram of high transition-
temperature superconductors remains enigmatic. A central issue is the origin of
the differences in the physical properties of these copper oxides doped to
opposite sides of the superconducting region. In the overdoped regime, the
material behaves as a reasonably conventional metal, with a large Fermi
surface. The underdoped regime, however, is highly anomalous and appears to
have no coherent Fermi surface, but only disconnected "Fermi arcs". The
fundamental question, then, is whether underdoped copper oxides have a Fermi
surface, and if so, whether it is topologically different from that seen in the
overdoped regime. Here we report the observation of quantum oscillations in the
electrical resistance of the oxygen-ordered copper oxide YBa2Cu3O6.5,
establishing the existence of a well-defined Fermi surface in the ground state
of underdoped copper oxides, once superconductivity is suppressed by a magnetic
field. The low oscillation frequency reveals a Fermi surface made of small
pockets, in contrast to the large cylinder characteristic of the overdoped
regime. Two possible interpretations are discussed: either a small pocket is
part of the band structure specific to YBa2Cu3O6.5 or small pockets arise from
a topological change at a critical point in the phase diagram. Our
understanding of high-transition temperature (high-Tc) superconductors will
depend critically on which of these two interpretations proves to be correct.
