Jet production in deep inelastic scattering for 120 < Q 2 < 3600 GeV 2 has been studied using data from an integrated luminosity of 3.2 pb - 1 collected with the ZEUS detector at HERA. Jets are identified with the JADE algorithm. A cut on the angular distribution of parton emission in the γ * -parton centre-of-mass system minimises the experimental and theoretical uncertainties in the determination of the jet rates. The jet rates, when compared to O(α s 2 ) perturbative QCD calculations, allow a precise determination of α s (Q) in three Q 2 -intervals. The values are consistent with a running of α s (Q), as expected from QCD. Extrapolating to Q = M Z 0 yields α s (M Z 0 ) = 0.117 +/- 0.005 (stat) + 0 . 0 0 4 - 0 . 0 0 5 (syst e x p ) +/- 0.007 (syst t h e o r y ).