  We report on the data set, data handling, and detailed analysis techniques of
the first neutrino-mass measurement by the Karlsruhe Tritium Neutrino (KATRIN)
experiment, which probes the absolute neutrino-mass scale via the $\beta$-decay
kinematics of molecular tritium. The source is highly pure, cryogenic T$_2$
gas. The $\beta$ electrons are guided along magnetic field lines toward a
high-resolution, integrating spectrometer for energy analysis. A silicon
detector counts $\beta$ electrons above the energy threshold of the
spectrometer, so that a scan of the thresholds produces a precise measurement
of the high-energy spectral tail. After detailed theoretical studies,
simulations, and commissioning measurements, extending from the molecular
final-state distribution to inelastic scattering in the source to subtleties of
the electromagnetic fields, our independent, blind analyses allow us to set an
upper limit of 1.1 eV on the neutrino-mass scale at a 90\% confidence level.
This first result, based on a few weeks of running at a reduced source
intensity and dominated by statistical uncertainty, improves on prior limits by
nearly a factor of two. This result establishes an analysis framework for
future KATRIN measurements, and provides important input to both particle
theory and cosmology.
