We report here the unexpected observation of significant room-temperature ferromagnetism in a semiconductor doped with nonmagnetic impurities, Cu-doped TiO₂ thin films grown by pulsed laser deposition. The magnetic moment, calculated from the magnetization curves, resulted surprisingly large, about 1.5 µB per Cu atom. A large magnetic moment was also obtained from ab initio calculations, but only if an oxygen vacancy in the nearest-neighbor shell of Cu was present. This result suggests that the role of oxygen vacancies is crucial for the appearance of ferromagnetism. The calculations also predict that Cu doping favors the formation of oxygen vacancies.