  We report on the \textit{Fermi}-LAT observations of the Geminga pulsar, the
second brightest non-variable GeV source in the $\gamma$-ray sky and the first
example of a radio-quiet $\gamma$-ray pulsar. The observations cover one year,
from the launch of the $Fermi$ satellite through 2009 June 15. A data sample of
over 60,000 photons enabled us to build a timing solution based solely on
$\gamma$ rays. Timing analysis shows two prominent peaks, separated by $\Delta
\phi$ = 0.497 $\pm$ 0.004 in phase, which narrow with increasing energy. Pulsed
$\gamma$ rays are observed beyond 18 GeV, precluding emission below 2.7 stellar
radii because of magnetic absorption. The phase-averaged spectrum was fitted
with a power law with exponential cut-off of spectral index $\Gamma$ = (1.30
$\pm$ 0.01 $\pm$ 0.04), cut-off energy $E_{0}$ = (2.46 $\pm$ 0.04 $\pm$ 0.17)
GeV and an integral photon flux above 0.1 GeV of (4.14 $\pm$ 0.02 $\pm$ 0.32)
$\times$ 10$^{-6}$ cm$^{-2}$ s$^{-1}$. The first uncertainties are statistical
and the second are systematic. The phase-resolved spectroscopy shows a clear
evolution of the spectral parameters, with the spectral index reaching a
minimum value just before the leading peak and the cut-off energy having maxima
around the peaks. Phase-resolved spectroscopy reveals that pulsar emission is
present at all rotational phases. The spectral shape, broad pulse profile, and
maximum photon energy favor the outer magnetospheric emission scenarios.

High Energy Astrophysical Phenomena