AbstractSince the discovery of neutrino oscillations, we know that neutrinos have non-zero mass. However, the absolute neutrino-mass scale remains unknown. Here we report the upper limits on effective electron anti-neutrino mass, m_ν, from the second physics run of the Karlsruhe Tritium Neutrino experiment. In this experiment, m_ν is probed via a high-precision measurement of the tritium β-decay spectrum close to its endpoint. This method is independent of any cosmological model and does not rely on assumptions whether the neutrino is a Dirac or Majorana particle. By increasing the source activity and reducing the background with respect to the first physics campaign, we reached a sensitivity on m_ν of 0.7 eV c^–2 at a 90% confidence level (CL). The best fit to the spectral data yields ${{\mbox{}}}{m}_{ν }^{2}{{\mbox{}}}$ m ν 2 = (0.26 ± 0.34) eV² c^–4, resulting in an upper limit of m_ν < 0.9 eV c^–2 at 90% CL. By combining this result with the first neutrino-mass campaign, we find an upper limit of m_ν < 0.8 eV c^–2 at 90% CL.