An on-demand single-photon source is a key element in a series of prospective quantum technologies and applications. Here we demonstrate the operation of a tuneable on-demand microwave photon source based on a fully controllable superconducting artificial atom strongly coupled to an open-ended transmission line. The atom emits a photon upon excitation by a short microwave π-pulse applied through a control line. The intrinsically limited device efficiency is estimated to be in the range 65–80% in a wide frequency range from 7.75 to 10.5 GHz continuously tuned by an external magnetic field. The actual demonstrated efficiency is also affected by the excited state preparation, which is about 90% in our experiments. The single-photon generation from the single-photon source is additionally confirmed by anti-bunching in the second-order correlation function. The source may have important applications in quantum communication, quantum information processing and sensing.