We exploited angle-resolved photoelectron spectroscopy (ARPES) with high energy resolution to map the band structure of the typical one-dimensional organic conductor TTF-TCNQ. We directly observe dispersing spectral features corresponding to the donor (TTF) and acceptor (TCNQ) bands, and their crossing at the Fermi wavevector k(F). Near the chemical potential, however, the quasiparticle peak are completely renormalized away, and the spectrum is totally incoherent. The suggest that we have observed the dispersion and spectral function of a metallic non-Fermi liquid.