Oxides possess several interesting properties, such as ferroelectricity, magnetism, superconductivity, and multiferroic behavior, which can effectively be used oxide electronics based on epitaxially grown heterostructures. The microscopic properties of oxide interfaces may have a strong impact on the electrical transport properties of these heterostructures. It was recently demonstrated that high electrical conductivity and mobility can be achieved in the system of an ultrathin $LaAlO_3$ film deposited on a $TiO_2$-terminated $SrTiO_3$ substrate, which was a remarkable result because the conducting layer was at the interface between two insulators. In this study, we observe that the current-voltage characteristics exhibit $LaAlO_3$ thickness dependence of electrical conductivity in $TiO_2$-terminated $SrTiO_3$. We find that the $LaAlO_3$ layers with a thickness of up 3 unit cells, result in highly insulating interfaces, whereas those with thickness of 4 unit cells and above result in conducting interfaces.