A precision measurement of jet cross sections in neutral current
deep-inelastic scattering for photon virtualities $5.5<\Qsq<80\,\GeVsq$ and
inelasticities $0.2<y<0.6$ is presented, using data taken with the H1
detector at HERA, corresponding to an integrated luminosity of
$290\,\invpb$.
Double-differential inclusive jet, dijet and trijet cross
sections are measured simultaneously and are presented as a function
of jet transverse momentum observables and as a function of \Qsq. Jet
cross sections normalised to the inclusive neutral current DIS cross
section in the respective \Qsq\ interval are also determined. Previous
results of inclusive jet cross sections in the range $150<\Qsq<15\,000\,\GeVsq$
are extended to low transverse jet jet momenta $5<\pt<7\,\GeV$. The data
are compared to predictions from perturbative QCD in next-to-leading
order in the strong coupling, in approximate next-to-next-to-leading
order and in full next-to-next-to-leading order. Using also the
recently published H1 jet data at high values of \Qsq, the strong
coupling constant \asmz\ is determined in next-to-leading order to
value of $\asmz=0.1172\,(4)_{\rm exp}\,(^{+53}_{-45})_{\rm th}$.