A precision measurement of jet cross sections in neutral current deep-inelastic scattering for photon virtualities 5.5 < Q 2 < 80 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 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 GeV 2 are extended to low transverse jet momenta 5 < P 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-nextto-leading order. Using also the recently published H1 jet data at high values of Q 2 , the strong coupling constant α s (M Z ) is determined in next-to-leading order.