We have investigated the magnetic properties of pure ZnO thin films grown under N₂ pressure on a-, c-, and r-plane Al₂O₃ substates by pulsed-laser deposition. The substrate temperature and the N₂ pressure were varied from room temperature to 570 °C and from 0.007 to 1.0 mbar, respectively. The magnetic properties of bare substates and ZnO films were investigated by SQUID magnetometry. ZnO films grown on c- and a-plane Al₂O₃ substrates did not show significant ferromagnetism. However, ZnO films grown on r-plane Al₂O₃ showed reproducible ferromagnetism at 300 K when grown at 300-400 °C and 0.1-1.0 mbar N₂ pressure. Positron annihilation spectroscopy measurements as well as density-functional theory calculations suggest that the ferromagnetism in ZnO films is related to Zn vacancies.