The concept of a next-generation spallation-driven ultracold neutron (UCN) source capable of delivering an integrated flux of ∼ 10 9 UCN s −1 is presented. A novel "inverse geometry" design is used with 40 liters of superfluid 4 He (He-II) as converter cooled with state-of-the-art sub-cooled cryogenic technology to ∼ 1.6 K. Our source design is optimized for a 100 W maximum thermal heat load constraint on the He-II and its vessel. In this paper, we first explore a modified Lujan-Center Mark-3 target for UCN production as a benchmark. We then present the baseline concept of our inverse geometry source design that gives a total UCN production rate in the converter of P UCN = 2.4 × 10 8 s −1 . In our inverse geometry, the spallation target is wrapped symmetrically around the cryogenic UCN converter to permit raster scanning the proton beam over a relatively large volume of tungsten spallation target to reduce the demand on the cooling requirements, which makes it reasonable to assume that water edge-cooling only is sufficient. Our design is refined in several steps to reach P UCN = 2.1 × 10 9 s −1 under our other restriction of 1 MW maximum available proton beam power. We then study effects of the He-II scattering kernel used as well as reductions in P UCN due to pressurization to reach a estimate of P UCN = 1.8 × 10 9 s −1 . Finally, we provide an estimate for the UCN extraction efficiency to show that the total extracted UCN rate out of the converter can be as large as R ex ≈ 6 × 10 8 s −1 for the He-II at 1.6 K out of a 18 cm diameter guide. The UCN extraction loss is dominated by upscattering in the He-II so that if the He-II can be cooled further to 1.4 K, R ex ≈ 8 × 10 8 s −1 can be attained. These extracted UCN rates are around an order of magnitude higher than the strongest proposed sources so far, and is around three orders of magnitude stronger than existing sources.He-II offers much higher thermal conductivities and a UCN upscattering time constant given by τ up ≈ (100 s K 7 )/T 7 (see Sec. VI), resulting in UCN loss times 3 s for a He-II bath at a temperature T = 1.6 K. At these temperatures, the total scattering mean-free-path for 8.9 Å (1 meV) neutrons, the primary UCN producing CN component, is ≈ 18 m. 25 This permits the design of a simple, large volume source where the converter material also serves as the coolant.There are two "in-pile" He-II sources currently under construction. One is at TRIUMF, Canada, where the source is coupled to a 500 MeV proton beam with 20 kW incident beam power. [26][27][28][29] The next generation version of this source will have an expected R ex ∼ 10 6 s −1 and UCN density deliverable to an experimental volume of ∼ 6, 000 cm −3 (both polarized).