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Voltage-controlled spin electronics is crucial for continued progress in information technology. It aims at reduced power consumption, increased integration density and enhanced functionality where non-volatile memory is combined with highspeed logical processing. Promising spintronic device concepts use the electric control of interface and surface magnetization. From the combination of magnetometry, spin-polarized photoemission spectroscopy, symmetry arguments and first-principles calculations, we show that the (0001) surface of magnetoelectric Cr2O3 has a roughness-insensitive, electrically switchable magnetization. Using a ferromagnetic Pd/Co multilayer deposited on the (0001) surface of a Cr2O3 single crystal, we achieve reversible, room-temperature isothermal switching of the exchange-bias field between positive and negative values by reversing the electric field while maintaining a permanent magnetic field. This effect reflects the switching of the bulk antiferromagnetic domain state and the interface magnetization coupled to it. The switchable exchange bias sets in exactly at the bulk Néel temperature.