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We propose a new method for production of ultracold neutrons (UCNs) in superfluid helium. The principal idea of the method consists of installing a helium UCN source into an external beam of thermal or cold neutrons and in surrounding this source with a solid methane moderator/reflector cooled down to the temperature of ~4 K. The moderator/reflector plays the role of an external source of cold neutrons, which are needed to produce UCNs. The flux of accumulated cold neutrons could exceed the flux of incident thermal or cold neutrons due to their numerous reflections from methane; also the source size could be significantly larger than the incident beam diameter. We provide preliminary calculations of cooling of neutrons in a cavity inside solid methane. These calculations show that such a source of UCNs, being installed at an intense source of thermal or cold neutrons like the ILL reactor in Grenoble, the PIK reactor in Gatchina or the ESS spallation source in Lund, could provide the UCN density of the order of 10^5 cm^{-3}, while the production rate would be of the order of 10^7 UCN/s^{-1}. Main advantages of such a UCN source include its low radiative and thermal load, relatively low cost and convenient accessibility for any maintenance work. We have carried out an experiment on cooling of thermal neutrons in a methane cavity with the wall temperature of 6.6 K. The experimental data confirm the results of our calculations of the spectrum and flux of neutrons in the methane cavity.