Ab initio studies of the single-photon ionization of LiH(X (1)Sigma(+)) leading to LiH+(X (2)Sigma(+)) are reported. The process is studied as function of the internuclear distance (2 a.u.-6 a.u.) and the kinetic energy of the electron (0.05 eV-10 eV). The calculations are based on the so-called iterative Schwinger approach which is implemented within the frozen-core single center approximation. By comparing vibrationally resolved cross sections calculated beyond and within the Franck-Condon principle we find, that the Franck-Condon principle applies relatively well to the present photoionization process. The vibrational wave functions needed for these calculations are obtained from very accurate CI-potentials, also reported in this paper. For LiH the first 10 vibrational spacings are found to agree with the experimental data within 0.2%. Photoelectron spectra (PES) calculated under simulated experimental conditions suggest that photoionization experiments on LiH may yield information on the vibrational structure of LiH+. The calculated permanent dipole moment of the LiH ground state shows strong influence of the ion-pair channel (Li+H-).