Al2O3 samples grown with natural well-ordered channels (having sizes of 30–90 nm in diameter) perpendicular to the surface, one side closed, were measured by means of Doppler broadening (DB) by variable energy positron (VEP) beam and positron annihilation lifetime (PAL) spectroscopy. The samples annealed at 300 and 560 °C showed amorphous structure and no positronium (Ps) formation. The sample annealed at 850 °C showed γ-phase polycrystalline structure and Ps formation. We were not able to prove from the PAL spectroscopy results themselves whether Ps is confined to the channels. We discuss the results of decomposition of the DB annihilation peak into four Gaussians. We conclude that the greater Ps fraction is confined to the channels. From the Compton-to-peak ratio analysis, the fraction of all the injected positrons which form o-Ps and in this form escape from the sample (for a VEP experiment, we have e+ injected from the open side of the channels) has been estimated as being ~20%.