Using three- and four-body decays of D mesons produced in semileptonic b-hadron decays, precision measurements of D meson mass differences are made together with a measurement of the D-0 mass. The measurements are based on a dataset corresponding to an integrated luminosity of 1.0 fb(-1) collected in pp collisions at 7 TeV. Using the decay D-0 -> K+K-K-pi(+), the D-0 mass is measured to be M(D-0) = 1864.75 +/- 0.15 (stat) +/- 0.11 (syst) MeV/c(2). The mass differences M(D+) - M(D-0) = 4.76 +/- 0.12 (stat) +/- 0.07 (syst) MeV/c(2), M(D-s(+)) - M(D+) = 98.68 +/- 0.03 (stat) +/- 0.04 (syst) MeV/c(2) are measured using the D-0 -> K+K-pi(+)pi(-) and D-(s)(+) -> K+K-pi(+) modes.