We propose a new method for production of ultracold neutrons (UCNs) in superfluid helium. The principal idea consists in 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 ~4 K. The moderator plays the role of an external source of cold neutrons needed to produce UCNs. The flux of accumulated neutrons could exceed the flux of incident 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. These calculations show that such a source being installed at an intense source of thermal or cold neutrons like the ILL or PIK reactor or the ESS spallation source could provide the UCN density 105 cm−3, the production rate 107 UCN/s−1. Main advantages of such an UCN source include its low radiative and thermal load, relatively low cost, and convenient accessibility for any maintenance. We have carried out an experiment on cooling of thermal neutrons in a methane cavity. The data confirm the results of our calculations of the spectrum and flux of neutrons in the methane cavity.