  Integrating multiple properties in a single system is crucial for the
continuous developments in electronic devices. However, some physical
properties are mutually exclusive in nature. Here, we report the coexistence of
two seemingly mutually exclusive properties-polarity and two-dimensional
conductivity-in ferroelectric Ba$_{0.2}$Sr$_{0.8}$TiO$_3$ thin films at the
LaAlO$_3$/Ba$_{0.2}$Sr$_{0.8}$TiO$_3$ interface at room temperature. The
polarity of a ~3.2 nm Ba$_{0.2}$Sr$_{0.8}$TiO$_3$ thin film is preserved with a
two-dimensional mobile carrier density of ~0.05 electron per unit cell. We show
that the electronic reconstruction resulting from the competition between the
built-in electric field of LaAlO$_3$ and the polarization of
Ba$_{0.2}$Sr$_{0.8}$TiO$_3$ is responsible for this unusual two-dimensional
conducting polar phase. The general concept of exploiting mutually exclusive
properties at oxide interfaces via electronic reconstruction may be applicable
to other strongly-correlated oxide interfaces, thus opening windows to new
functional nanoscale materials for applications in novel nanoelectronics.
