Given the big cross section for t \bar t production in proton-proton collisions at 14 TeV, the LHC with its high luminosity will be, among others, a top factory, allowing a precision measurement of the top quark mass. Based on a detailed simulation of the CMS detector, the following top mass reconstruction accuracies are possible in the respective final states with the present knowledge of experimental and theoretical uncertainties. \Delta m_t({dileptonic}, 1 fb^-1) = \pm 1.5 (stat) \pm 2.9 {(syst) GeV}/c²

\Delta m_t({semileptonic}, 1 fb^-1) = \pm 0.7 (stat) \pm 1.9 {(syst) GeV}/c²

\Delta m_t({fully hadronic}, 1 fb^-1)= \pm 0.6 (stat) \pm 4.2 {(syst) GeV}/c²

\Delta m_t({dileptonic}, 10 fb^-1) = \pm 0.5 (stat) \pm 1.1 {(syst) GeV}/c²

 \Delta m_t({semileptonic}, 10 fb^-1) = \pm 0.2 (stat) \pm 1.1 {(syst) GeV}/c²

\Delta m_t(J/\Psi , 20 fb^-1) = \pm 1.2 (stat) \pm 1.5 {(syst) GeV}/c²

A combined top mass accuracy of {O}(1 GeV/c²) for 10- 20 fb^-1 of well-understood CMS data will be feasible.

PACS numbers: 14.65.Ha