Single-photon generation is an important proof of the underlying quantum nature of a physical process and a ubiquitous tool for scientific exploration, with applications ranging from spectroscopy and metrology to quantum computing. In the microwave regime, emission of antibunched radiation has so far relied on coherent control of Josephson qubits requiring precisely calibrated microwave pulses. In this work, we experimentally demonstrate the operation of a bright on-demand source of quantum microwave radiation driven by a simple dc voltage bias across a Josephson junction. Our source is based on photon emission into a microwave resonator through inelastic Cooper pair tunneling regulated by a high-impedance circuit preventing simultaneous tunnel events. It is characterized by its normalized second-order correlation function of corresponding to antibunching in the single-photon regime. Our device can be triggered, and its in situ tunable emission rate exceeds those obtained with current microwave single-photon sources by more than 1 order of magnitude.