  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
$\gamma^*$-parton centre-of-mass system minimises the experimental and
theoretical uncertainties in the determination of the jet rates. The jet rates,
when compared to ${\cal O}$($\alpha_{s}$^2$) perturbative QCD calculations,
allow a precise determination of $\alpha_{s}(Q)$ in three $Q^2$-intervals. The
values are consistent with a running of $\alpha_{s}(Q)$, as expected from QCD.
Extrapolating to $Q=M_{Z^0}$ yields $\alpha_{s}(M_{Z^0}) =
0.117~\pm~0.005~(stat)~^{+0.004}_{-0.005}~(syst_{exp})~
~{\pm~0.007}~(syst_{theory})$.
