Technically useful bulk superconductors must have high transport critical current densities, J_c, at operating temperatures. They also require a normal metal cladding to provide parallel electrical conduction, thermal stabilization, and mechanical protection of the generally brittle superconductor cores. The recent discovery of superconductivity at 39 K in magnesium diboride (MgB₂) presents a new possibility for significant bulk applications^,,,, but many critical issues relevant for practical wires remain unresolved. In particular, MgB₂ is mechanically hard and brittle and therefore not amenable to drawing into the desired fine-wire geometry. Even the synthesis of moderately dense, bulk MgB₂ attaining 39 K superconductivity is a challenge because of the volatility and reactivity of magnesium. Here we report the successful fabrication of dense, metal-clad superconducting MgB₂ wires, and demonstrate a transport J_c in excess of 85,000 A cm^-2 at 4.2 K. Our iron-clad fabrication technique takes place at ambient pressure, yet produces dense MgB₂ with little loss of stoichiometry. While searching for a suitable cladding material, we found that other materials dramatically reduced the critical current, showing that although MgB₂ itself does not show the ‘weak-link’ effect characteristic of the high-T_c superconductors, contamination does result in weak-link-like behaviour.