$MgB_2$ becomes superconducting just below 40 K. Whereas porous polycrystalline samples of $MgB_2$ can be synthesized from boron powders, in this letter we demonstrate that dense wires of $MgB_2$ can be prepared by exposing boron filaments to $Mg$ vapor. The resulting wires have a diameter of 160 ${mu}m$, are better than 80% dense and manifest the full $chi = -1/4{pi}$ shielding in the superconducting state. Temperature-dependent resistivity measurements indicate that $MgB_2$ is a highly conducting metal in the normal state with $rho (40 K)$ = 0.38 $mu Ohm$-$cm$. Using this value, an electronic mean free path, $l approx 600~AA$ can be estimated, indicating that $MgB_2$ wires are well within the clean limit. $T_c$, $H_{c2}(T)$, and $J_c$ data indicate that $MgB_2$ manifests comparable or better superconducting properties in dense wire form than it manifests as a sintered pellet.
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