Inclusive jet production is studied in neutral current deep-inelastic positron–proton scattering at large four momentum transfer squared with the H1 detector at HERA. Single and double differential inclusive jet cross sections are measured as a function of Q2 and of the transverse energy ET of the jets in the Breit frame. The measurements are found to be well described by calculations at next-to-leading order in perturbative QCD. The running of the strong coupling is demonstrated and the value of αs(MZ) is determined. The ratio of the inclusive jet cross section to the inclusive neutral current cross section is also measured and used to extract a precise value for .