In this Letter we report the first results on π(±), K(±), p, and p production at midrapidity (|y|<0.5) in central Pb-Pb collisions at sqrt[s(NN)] = 2.76 TeV, measured by the ALICE experiment at the LHC. The p(T) distributions and yields are compared to previous results at sqrt[s(NN)] = 200 GeV and expectations from hydrodynamic and thermal models. The spectral shapes indicate a strong increase of the radial flow velocity with sqrt[s(NN)], which in hydrodynamic models is expected as a consequence of the increasing particle density. While the K/π ratio is in line with predictions from the thermal model, the p/π ratio is found to be lower by a factor of about 1.5. This deviation from thermal model expectations is still to be understood.