Ultracold neutrons (UCNs) play an important role for precise measurements of the properties of the neutron and its interactions. During the past 25 years, a neutron turbine coupled to a liquid deuterium cold neutron source at a high-flux reactor has defined the state of the art for UCN production, despite a long history of efforts towards a new generation of UCN sources. This Letter reports a world-best UCN density available for users, achieved with a new source based on conversion of cold neutrons in superfluid helium. A conversion volume of 5 liters provides at least 274,000 UCN in a single accumulation run. Cyclically repeated operation of the source has been demonstrated, as well.