  Superconductivity of YBa$_2$Cu$_3$O$_{7-\delta}$ single crystals was
investigated in small magnetic fields. In magnetic measurements the
superconducting transition for $\textbf{H} \| c$ appears 0.4 K higher than for
$\textbf{H} \bot c$. 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 \textit{ab}-plane and along the \textit{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 $T_c$, the mean-field transition temperature.

Superconductivity