Using density-functional studies, we show that a spin-orbital entangled two-dimensional electron gas (2DEG) forms at the (001) LaAlO3/Sr2IrO4 polar interface between an ordinary band insulator and a spin-orbit coupled Mott insulator, aided by the combination of the spin-orbit coupling, Coulomb interaction, and polar catastrophe. Quite remarkably, the 2DEG is found to be localized on a single IrO2 monolayer, unlike other polar interfaces such as LaAlO3/SrTiO3, where the 2DEG is several monolayers thick. The electron gas occupies the upper J eff =1/2 Hubbard band in the interface layer, which becomes half-filled with a simple square-like Fermi surface. If successfully grown, this would be the first candidate material to host the spin-orbital entangled 2DEG. 75.70.Tj