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The production of Ξ(1321)− and Ξ¯¯¯¯(1321)+ hyperons in inelastic p+p interactions is studied in a fixed target experiment at a beam momentum of 158 GeV/c. Double differential distributions in rapidity y and transverse momentum pT are obtained from a sample of 33M inelastic events. They allow to extrapolate the spectra to full phase space and to determine the mean multiplicity of both Ξ− and Ξ¯¯¯¯+. The rapidity and transverse momentum spectra are compared to transport model predictions. The Ξ− mean multiplicity in inelastic p+p interactions at 158 GeV/c is used to quantify the strangeness enhancement in A+A collisions at the same centre-of-mass energy per nucleon pair.