Microwaves have widespread use in classical communication technologies, from long-distance broadcasts to short-distance signals within a computer chip. Like all forms of light, microwaves, even those guided by the wires of an integrated circuit, consist of discrete photons. To enable quantum communication between distant parts of a quantum computer, the signals must also be quantum, consisting of single photons, for example. However, conventional sources can generate only classical light, not single photons. One way to realize a single-photon source is to collect the fluorescence of a single atom. Early experiments measured the quantum nature of continuous radiation, and further advances allowed triggered sources of photons on demand. To allow efficient photon collection, emitters are typically placed inside optical or microwave cavities, but these sources are difficult to employ for quantum communication on wires within an integrated circuit. Here we demonstrate an on-chip, on-demand single-photon source, where the microwave photons are injected into a wire with high efficiency and spectral purity. This is accomplished in a circuit quantum electrodynamics architecture, with a microwave transmission line cavity that enhances the spontaneous emission of a single superconducting qubit. When the qubit spontaneously emits, the generated photon acts as a flying qubit, transmitting the quantum information across a chip. We perform tomography of both the qubit and the emitted photons, clearly showing that both the quantum phase and amplitude are transferred during the emission. Both the average power and voltage of the photon source are characterized to verify performance of the system. This single-photon source is an important addition to a rapidly growing toolbox for quantum optics on a chip.