  We report the results of the second measurement campaign of the Karlsruhe
Tritium Neutrino (KATRIN) experiment. KATRIN probes the effective electron
anti-neutrino mass, $m_{\nu}$, via a high-precision measurement of the tritium
$\beta$-decay spectrum close to its endpoint at $18.6\,\mathrm{keV}$. In the
second physics run presented here, the source activity was increased by a
factor of 3.8 and the background was reduced by $25\,\%$ with respect to the
first campaign. A sensitivity on $m_{\nu}$ of $0.7\,\mathrm{eV/c^2}$ at
$90\,\%$ confidence level (CL) was reached. This is the first sub-eV
sensitivity from a direct neutrino-mass experiment. The best fit to the
spectral data yields $m_{\nu}^2 = (0.26\pm0.34)\,\mathrm{eV^4/c^4}$, resulting
in an upper limit of $m_{\nu}<0.9\,\mathrm{eV/c^2}$ ($90\,\%$ CL). By combining
this result with the first neutrino mass campaign, we find an upper limit of
$m_{\nu}<0.8\,\mathrm{eV/c^2}$ ($90\,\%$ CL).
