We present a systematic study of the transmission of low-energy (10 eV) ${\mathrm{O}}^{+}$ ions through ultrathin films of Ar, Kr, and Xe. The ions are produced by electron-stimulated desorption from an oxidized W(100) crystal; they desorb from the surface in directions close to the surface normal with a peak kinetic energy of \ensuremath{\sim}7 eV and their yield, mass/energy, and angle are measured with a digital electron-stimulated desorption ion angular distribution (ESDIAD) detector. Rare gases are condensed at \ensuremath{\sim}25 K onto the oxidized W(100) crystal and their film thickness is determined by means of thermal-desorption spectroscopy. The ${\mathrm{O}}^{+}$ ions desorbed in the presence of a rare-gas film have to pass through the film before reaching the detector. We find that 10% of ${\mathrm{O}}^{+}$ can be transmitted through 1.6 atomic layers of Ar, 2.9 ML of Kr, and 4.0 ML of Xe. From the ${\mathrm{O}}^{+}$ signal attenuation by films thicker than 2 ML we derive attenuation cross sections of 6.0\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}15}$ ${\mathrm{cm}}^{2}$ for Ar, 2.2\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}15}$ ${\mathrm{cm}}^{2}$ for Kr, and 1.5\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}15}$ ${\mathrm{cm}}^{2}$ for Xe. For Xe, we observe indications that the angular distribution of the ions changes due to large-angle scattering, and for Kr (and previously for Xe) we measure a shift in the energy distribution towards lower energies; we interpret this to be due to elastic forward scattering of the oxygen ions by the Xe atoms.We attribute the attenuation of the ${\mathrm{O}}^{+}$ in the films mainly to elastic backscattering; we suggest that either a high neutralization probability of ${\mathrm{O}}^{+}$ in the Ar film (charge transfer) or an Ar structure different from fcc (such as blocking of ${\mathrm{O}}^{+}$ by Ar) is the reason for the strong attenuation of ${\mathrm{O}}^{+}$ in Ar. We find greater attenuation per monolayer for thicker films than for the first monolayer; we correlate this with the fcc structure of the rare-gas films. We discuss the energy loss of the primary electrons in the rare-gas film, the effect of the adsorption of rare gases on the electron-stimulated desorption process, and the possibility of preferential ${\mathrm{O}}^{+}$ desorption through channels in the rare-gas film. We draw conclusions from our results concerning the depth of origin of secondary ions desorbed under the influence of electron, photon, or ion radiation.