A precision measurement of jet cross sections in neutral current deep-inelastic scattering for photon virtualities 5.5 < Q(2) < 80 GeV2 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 GeV2 are extended to low transverse jet momenta 5 < P-T(jet) < 7GeV. 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 as (MZ) is determined in next-to-leading order.