The production of D*± (2010) mesons in deep inelastic scattering has been measured in the ZEUS detector at HERA using an integrated luminosity of 37 pb-1. The decay channels D*+ → D0π+ (+c.c.), with D0 → K-π+ or D0 → K-π-π+π+, have been used to identify the D mesons. The e+p cross section for inclusive D*± production with 1 < Q2 < 600 GeV2 and 0.02 < y < 0.7 is 8.31±0.31(stat.)+0.30-0.50 (syst.) nb in the kinematic region 1.5 < pT(D*±) < 15 GeV and \η(D*±)\ < 1.5. Differential cross sections are consistent with a next-to-leading-order perturbative-QCD calculation when using charm-fragmentation models which take into account the interaction of the charm quark with the proton remnant. The observed cross section is extrapolated to the full kinematic region in pT(D*±) and η(D*±) in order to determine the charm contribution, Fcc̄2 (x, Q2), to the proton structure function. The ratio Fcc̄2/F2 rises from ≃10% at Q2 ≃ 1.8GeV2 to ≃30% at Q2 ≃130GeV2 for x values in the range 10-4 to 10-3.