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 $Q^2 < 1$ GeV$^2$, and in deep inelastic scattering, where $2 < Q^2 < 100$ GeV$^2$.