We studied electronic structure of one-dimensional nanographite ribbons with a pair of zigzag edges by performing first-principles calculations within the local density approximation. The edge state of zigzag ribbons, which was predicted by the previous tight-binding calculations, was well reproduced by the present first-principles scheme. Nearly flat bands of the edge state were seen in zigzag ribbons which are condensed in a manner of AB-stacking as in bulk graphite. We took account of dangling bonds at edge sites and confirmed that the dangling-bond state near the Fermi level does not affect the nearly flat dispersion of the edge state. We also examined electronic structure of zigzag ribbons made up of an odd number of C-C chains and discussed the possibility of lattice distortion.