An exact expression for the Green's function G(k,omega) of one hole in the U --> infinity one-dimensional Hubbard model is derived from the known Bethe-ansatz solution of the model. Due to the spin-charge decoupling of the one-hole excitation spectrum and the Fermi-like statistics of the spin excitations, G shows branch-cut singularities at omega=+/-2 sink and a nontrivial dependence on the momentum of the hole. As a result, the hole can propagate although the quasiparticle weight vanishes.
