We report the results of the second measurement campaign of the Karlsruhe Tritium Neutrino (KATRIN) experiment. KATRIN probes the effective electron anti-neutrino mass, $m_{ u}$, 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 $m_{ u}$ 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 $m_{ u}^2 = (0.26pm0.34),mathrm{eV^4/c^4}$, resulting in an upper limit of $m_{ u}<0.9,mathrm{eV/c^2}$ ($90,%$ CL). By combining this result with the first neutrino mass campaign, we find an upper limit of $m_{ u}<0.8,mathrm{eV/c^2}$ ($90,%$ CL).