A measurement is presented of the charged hadron multiplicity in hadronic PbPb collisions, as a function of pseudorapidity and centrality, at a collision energy of 2.76 TeV per nucleon pair. The data sample is collected using the CMS detector and a minimum-bias trigger, with the CMS solenoid off. The number of charged hadrons is measured both by counting the number of reconstructed particle hits and by forming hit doublets of pairs of layers in the pixel detector. The two methods give consistent results. The charged hadron multiplicity density, dN(ch)/d eta vertical bar(eta=0), for head-on collisions is found to be 1612 +/- 55, where the uncertainty is dominated by systematic effects. Comparisons of these results to previous measurements and to various models are also presented.