The top-quark pair production cross section in 7 TeV center-of-mass energy proton-proton collisions is measured using data collected by the CMS detector at the LHC. The measurement uses events with one jet identified as a hadronically decaying tau lepton and at least four additional energetic jets, at least one of which is identified as coming from a b quark. The analyzed data sample corresponds to an integrated luminosity of 3.9 inverse femtobarns recorded by a dedicated multijet plus hadronically decaying tau trigger. A neural network has been developed to separate the top-quark pairs from the W+jets and multijet backgrounds. The measured value of sigma(ttbar) = 152 +/- 12 (stat.) +/- 32 (syst.) +/- 3 (lum.) pb is consistent with the standard model predictions.