We report the observation of a dense triangular network of one-dimensional (1D) metallic modes in a continuous and uniform monolayer of MoSe(2) grown by molecular-beam epitaxy. High-resolution transmission electron microscopy and scanning tunneling microscopy and spectroscopy studies show that these 1D modes are midgap states at inversion domain boundaries. Scanning tunneling microscopy and spectroscopy measurements further reveal intensity undulations of the metallic modes, presumably arising from the superlattice potentials due to the moiré pattern and the quantum confinement effect. A dense network of the metallic modes with a high density of states is of great potential for heterocatalysis applications. The interconnection of such midgap 1D conducting channels may also imply new transport behaviors distinct from the 2D bulk.