  We examine the single-particle excitation spectrum in the one-dimensional
Hubbard-Holstein model at half-filling by performing the dynamical density
matrix renormalization group (DDMRG) calculation. The DDMRG results are
interpreted as superposition of spectra for a spinless carrier dressed with
phonons. The superposition is a consequence of robustness of the spin-charge
separation against electron-phonon coupling. The separation is in contrast to
the coupling between phonon and spin degrees of freedom in two-dimensional
systems. We discuss implication of the results of the recent angle-resolved
photoemission spectroscopy measurements on SrCuO${}_{2}$.
