More than Moore'(1) captures a concept for overcoming limitations in silicon electronics by incorporating new functionalities in the constituent materials. Perovskite oxides are candidates because of their vast array of physical properties in a common structure. They also enable new electronic devices based on strongly-correlated electrons(2). The field effect transistor(3) and its derivatives have been the principal oxide devices investigated thus far(4-6), but another option is available in a different geometry: if the current is perpendicular to the interface, the strong internal electric fields generated at back-to-back heterojunctions can be used for oxide electronics, analogous to bipolar transistors(7). Here we demonstrate a perovskite heteroepitaxial metal-base transistor(8) operating at room temperature, enabled by interface dipole engineering(9). Analysis of many devices quantifies the evolution from hot-electron(10) to permeable-base(11) behaviour. This device provides a platform for incorporating the exotic ground states of perovskite oxides, as well as novel electronic phases at their interfaces(12-15).