THE recent discovery(1,2) of a family of mercury-based copper oxide superconductors having transition temperatures above 130 K is of considerable technological interest, But the viability of high-temperature superconductors for many applications will ultimately depend on the size of the current density, J(c), that they are able to support, not only at high temperatures, but also in high magnetic fields, For the cuprate superconductors, and in particular for Hg-based materials, the combination of high transition temperature(1-3) and large mass anisotropy implies that the transport properties will be intrinsically limited by large thermal fluctuations and short superconducting coherence lengths(4). Here we report that high-quality c-axis-oriented epitaxial films of the compound HgBa2CaCu6O6+delta (Hg-1212; ref, 5) can support large in-plane current densities at temperatures higher than has been achieved for other superconductors. In low magnetic fields oriented normal to the film surface, we find J(c) greater than or similar to 10(7) A cm(-2) at 5 K and J(c) similar to 10(5) A cm(-2) at 110 K, at least an order of magnitude larger than for Bi- or Tl-based films(6-11). For in-plane magnetic fields, the critical current (similar to 10(8) A cm(-2)) is close to the theoretical limit even at high fields, indicative of strong intrinsic pinning in this compound.