High-current applications of high-temperature superconductor form parts (e.g. plates, rods, tubes) need connections to conventional (copper) current leads of resistivities below 10 mu Omega cm². Bismuth-based BSCCO 2212 parts can be contacted by two different methods: (i) the conventional covering of the surface with gold or silver by sputtering or by vapour deposition, which always needs a post-deposition thermal treatment, and (ii) applying silver sheet to the superconductor part during its fabrication by a melt casting process. In both cases resistivities of 1-3 mu Omega cm² at 77 K could be achieved. Experiments and theoretical considerations to localize the partial resistances within the connection demonstrate that the interfacial resistivity of melt processed contacts, as well as sputtered or vapour deposited contacts after heat treatment, are lower by about one order of magnitude than the resistivity of the total connection.