The absolute production rate of ultracold neutrons (UCN) produced by the interaction of a cold neutron beam with superfluid helium has been measured over an incident energy range of 0.7 to 4 meV. The neutrons are reduced in energy to become UCN by creating phonon(s) in the superfluid. The separate roles played by single and multi-phonon emission processes have been identified. Detection and identification of UCN, those neutrons with energies less than ~250 neV and which can be stored in material bottles, were carried out using solid-state silicon detectors set within the superfluid helium. With a cold neutron flux of 2.62x10 7 neutrons cm - 2 s - 1 Å - 1 at 8.9 Å in the superfluid, the single-phonon production rate of UCN was measured to be (0.91+/-0.13) cm - 3 s - 1 , a value close to theoretical prediction. Multi-phonon emission processes for UCN production by higher energy neutrons were also observed and, in the beam used for this work at ILL, they contributed (24+/-2)% to the overall UCN production rate.