A precision measurement of jet cross sections in neutral current deep-inelastic scattering for photon virtualities Formula omitted and inelasticities Formula omitted is presented, using data taken with the H1 detector at HERA, corresponding to an integrated luminosity of Formula omitted. 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 Formula omitted. Jet cross sections normalised to the inclusive neutral current DIS cross section in the respective Formula omitted-interval are also determined. Previous results of inclusive jet cross sections in the range Formula omitted are extended to low transverse jet momenta Formula omitted. 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 Formula omitted, the strong coupling constant Formula omitted is determined in next-to-leading order.