The production of prompt charmed mesons D0, D+ and D∗+, and their antiparticles, was measured with the ALICE detector in Pb-Pb collisions at the centre-of-mass energy per nucleon pair, (Formula presented.) , of 2.76 TeV. The production yields for rapidity |y| < 0.5 are presented as a function of transverse momentum, pT, in the interval 1–36 GeV/c for the centrality class 0–10% and in the interval 1–16 GeV/c for the centrality class 30–50%. The nuclear modification factor RAA was computed using a proton-proton reference at (Formula presented.) TeV, based on measurements at (Formula presented.) TeV and on theoretical calculations. A maximum suppression by a factor of 5-6 with respect to binary-scaled pp yields is observed for the most central collisions at pT of about 10 GeV/c. A suppression by a factor of about 2-3 persists at the highest pT covered by the measurements. At low pT (1-3 GeV/c), the RAA has large uncertainties that span the range 0.35 (factor of about 3 suppression) to 1 (no suppression). In all pT intervals, the RAA is larger in the 30-50% centrality class compared to central collisions. The D-meson RAA is also compared with that of charged pions and, at large pT, charged hadrons, and with model calculations.