The first measurements of the muon-induced neutron flux at the Boulby Underground Laboratory are presented. The experiment was carried out with an 0.73 tonne liquid scintillation detector that also served as an anticoincidence system for the ZEPLIN-II direct dark matter search. The experimental method exploited the delayed coincidences between high-energy muon signals and gamma-rays from radiative neutron capture on hydrogen or other elements. The muon-induced neutron rate, defined as the average number of detected neutrons per detected muon, was measured as 0.079 ±0.003 (stat.) neutrons/muon using neutron-capture signals above 0.55 MeV in a time window of 40-190 μs after the muon trigger. Accurate Monte Carlo simulations of the neutron production, transport and detection in a precisely modeled laboratory and experimental setup using the GEANT4 toolkit gave a result 1.8 times higher than the measured value. The difference greatly exceeds all statistical and systematic uncertainties. As the vast majority of neutrons detected in the current setup were produced in lead we evaluated from our measurements the neutron yield in lead as (1.31 ±0.06) × 10-3 neutrons/muon/(g/cm2) for a mean muon energy of about 260 Ge V.