A measurement of the beauty production cross section in ep collisions at a centre-of-mass energy of 319 GeV is presented. The data were collected with the H1 detector at the HERA collider in the years 1999–2000. Events are selected by requiring the presence of jets and muons in the final state. Both the long lifetime and the large mass of b-flavoured hadrons are exploited to identify events containing beauty quarks. Differential cross sections are measured in photoproduction, with photon virtualities Q2 < 1 GeV2, and in deep inelastic scattering, where 2 < Q2 < 100 GeV2. The results are compared with perturbative QCD calculations to leading and next-to-leading order. The predictions are found to be somewhat lower than the data.