Spectroscopic ellipsometry was used to study the dielectric function of LiCuVO4, a compound comprised of chains of edge-sharing CuO4 plaquettes, in the spectral range 0.75-6.5 eV at temperatures 7-300 K. For photon polarization along the chains, the data reveal a weak but well-resolved two-peak structure centered at 2.15 and 2.95 eV whose spectral weight is strongly enhanced upon cooling near the magnetic ordering temperature. We identify these features as an exciton doublet in the Mott-Hubbard gap that emerges as a consequence of the Coulomb interaction between electrons on nearest and next-nearest-neighbor sites along the chains. Our results and methodology can be used to address the role of the long-range Coulomb repulsion for compounds with doped copper-oxide chains and planes.