The production of $\Xi (1321)^{-}$ and $\overline{\Xi }(1321)^{+}$ hyperons in inelastic p+p interactions is studied in a fixed target experiment at a beam momentum of 158 $\hbox {Ge}\hbox {V}\!/\!c$. Double differential distributions in rapidity ${y}$ and transverse momentum $p_{T}$ 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 ${\Xi }{^-} $ and $\overline{\Xi }{^+} $. The rapidity and transverse momentum spectra are compared to transport model predictions. The ${\Xi }{^-} $ mean multiplicity in inelastic p+p interactions at 158 $\hbox {Ge}\hbox {V}\!/\!c$ is used to quantify the strangeness enhancement in A+A collisions at the same centre-of-mass energy per nucleon pair.