A search for narrow resonances with a mass of at least 1 TeV in the dijet mass spectrum is performed using pp collisions at sqrt(s)=7 TeV corresponding to an integrated luminosity of 1 inverse femtobarn, collected by the CMS experiment at the LHC. No resonances are observed. Upper limits at the 95% confidence level are presented on the product of the resonance cross section, branching fraction into dijets, and acceptance, separately for decays into quark-quark, quark-gluon, and gluon-gluon pairs. The data exclude new particles predicted in the following models at the 95% confidence level: string resonances with mass less than 4.00 TeV, E6 diquarks with mass less than 3.52 TeV, excited quarks with mass less than 2.49 TeV, axigluons and colorons with mass less than 2.47 TeV, and W' bosons with mass less than 1.51 TeV.