Measurements on nanoscale structures constructed from high-temperature superconductors are expected to shed light on the origin of superconductivity in these materials. To date, loops made from these compounds have had sizes of the order of hundreds of nanometres(8-11). Here, we report the results of measurements on loops of La(1.84)Sr(0.16)CuO(4), a high-temperature superconductor that loses its resistance to electric currents when cooled below approximately 38 K, with dimensions down to tens of nanometres. We observe oscillations in the resistance of the loops as a function of the magnetic flux through the loops. The oscillations have a period of h/2e, and their amplitude is much larger than the amplitude of the resistance oscillations expected from the Little-Parks effect. Moreover, unlike Little-Parks oscillations, which are caused by periodic changes in the superconducting transition temperature, the oscillations we observe are caused by periodic changes in the interaction between thermally excited moving vortices and the oscillating persistent current induced in the loops. However, despite the enhanced amplitude of these oscillations, we have not detected oscillations with a period of h/e, as recently predicted for nanoscale loops of superconductors with d-wave symmetry, or with a period of h/4e, as predicted for superconductors that exhibit stripes.