In the pseudogap state of the high-transition-temperature (high-Tc) copper oxide superconductors1, angle-resolved photoemission (ARPES) measurements have seen Fermi arcs—that is, open-ended gapless sections in the large Fermi surface2, 3, 4, 5, 6, 7, 8—rather than a closed loop expected of an ordinary metal. This is all the more puzzling because Fermi pockets (small closed Fermi surface features) have been suggested by recent quantum oscillation measurements9, 10, 11, 12, 13, 14. The Fermi arcs cannot be understood in terms of existing theories, although there is a solution in the form of conventional Fermi surface pockets associated with competing order, but with a back side that is for detailed reasons invisible to photoemission probes15. Here we report ARPES measurements of Bi2Sr2-xLaxCuO6+ (La-Bi2201) that reveal Fermi pockets. The charge carriers in the pockets are holes, and the pockets show an unusual dependence on doping: they exist in underdoped but not overdoped samples. A surprise is that these Fermi pockets appear to coexist with the Fermi arcs. This coexistence has not been expected theoretically.