The inclusive production of D ^∗±(2010) mesons in deep-inelastic scattering is studied with the H1 detector at HERA. In the kinematic region 1<Q²<100 GeV² and 0.05<y<0.7 an e⁺p cross section for inclusive D ^∗± meson production of 8.50±0.42(stat.)^+1.21_−1.00(syst.) nb is measured in the visible range p _tD^∗>1.5  GeV and |η _D^∗|<1.5 . Single and double differential inclusive D ^∗± meson cross sections are compared to perturbative QCD calculations in two different evolution schemes. The charm contribution to the proton structure, F₂^c(x,Q²), is determined by extrapolating the visible charm cross section to the full phase space. This contribution is found to rise from about 10% at Q²=1.5 GeV² to more than 25% at Q²=60 GeV² corresponding to x values ranging from 5×10⁻⁵ to 3×10⁻³.