The discovery of a two-dimensional electron gas (2DEG) at the LaAlO3/SrTiO3 interface(1) has resulted in the observation of many properties(2-5) not present in conventional semiconductor heterostructures, and so become a focal point for device applications(6-8). Its counterpart, the two-dimensional hole gas (2DHG), is expected to complement the 2DEG. However, although the 2DEG has been widely observed(9), the 2DHG has proved elusive. Herein we demonstrate a highly mobile 2DHG in epitaxially grown SrTiO3/LaAlO3/SrTiO3 heterostructures. Using electrical transport measurements and in-line electron holography, we provide direct evidence of a 2DHG that coexists with a 2DEG at complementary heterointerfaces in the same structure. First-principles calculations, coherent Bragg rod analysis and depth-resolved cathodoluminescence spectroscopy consistently support our finding that to eliminate ionic point defects is key to realizing a 2DHG. The coexistence of a 2DEG and a 2DHG in a single oxide heterostructure provides a platform for the exciting physics of confined electron-hole systems and for developing applications.