Superconductivity of YBa2Cu3O 7−δ single crystals was investigated in small magnetic fields. In magnetic measurements the superconducting transition for H c appears 0.4 K higher than for H⊥c. In this temperature range superconductivity is two-dimensional and the total thickness of superconducting layers is about 0.83 of the sample thickness, which is a consequence of the occurrence of the quasi-insulating plane in the unit cell of the crystal structure. Resistivity in the ab-plane and along the c-axis was measured simultaneously. In these measurements two-dimensional superconductivity was observed in a temperature range of 0.6-0.8 K with the clear signs of the Berezinskii-Kosterlitz-Touless (BKT) transition which occurs approximately 0.15 K below Tc, the mean-field transition temperature.