Temperature dependence of the penetration of magnetic flux into YBa 2 Cu 3 O 7Ϫ␦ disk-shaped thin films was investigated at low magnetic fields. We studied the conditions characteristic of complete and incomplete flux-penetration states. The experimental procedure involved the measurements of the profiles of trapped magnetic flux as a function of temperature and the decays of trapped flux at various points across the disk. We determined temperatures and magnetic fields at which the crossover between incomplete and complete flux penetration occurs. This crossover is governed by the temperature dependence of the critical current density. The distributions of the critical currents across the disk in the complete flux-penetration state were found to be temperature independent. No evidence for the Bean-Livingston surface barriers has been found.