  A precision measurement of jet cross sections in neutral current
deep-inelastic scattering for photon virtualities $5.5<Q^2<80\,{\rm GeV}^2$ and
inelasticities $0.2<y<0.6$ is presented, using data taken with the H1 detector
at HERA, corresponding to an integrated luminosity of $290\,{\rm pb}^{-1}$.
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 $Q^2$. Jet cross sections normalised to the
inclusive neutral current DIS cross section in the respective $Q^2$-interval
are also determined. Previous results of inclusive jet cross sections in the
range $150<Q^2<15\,000\,{\rm GeV}^2$ are extended to low transverse jet momenta
$5<P_{T}^{\rm jet}<7\,{\rm GeV}$. 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 $Q^2$,
the strong coupling constant $\alpha_s(M_Z)$ is determined in next-to-leading
order.
