  We have obtained a firm detection of Cyg X-1 during its hard and intermediate
spectral states in the energy range of 40 MeV--60 GeV based on observations by
the Fermi Large Area Telescope, confirming the independent results at
$\geq$60MeV of a previous work. The detection significance is $\simeq\!8\sigma$
in the 0.1--10 GeV range. In the soft state, we have found only upper limits on
the emission at energies $\gtrsim$0.1 MeV. However, we have found emission with
a very soft spectrum in the 40--80 MeV range, not detected previously. This is
likely to represent the high-energy cutoff of the high-energy power-law tail
observed in the soft state. Similarly, we have detected a $\gamma$-ray soft
excess in the hard state, which appears to be of similar origin. We have also
confirmed the presence of an orbital modulation of the detected emission in the
hard state, expected if the $\gamma$-rays are from Compton upscattering of
stellar blackbody photons. However, the observed modulation is significantly
weaker than that predicted if the blackbody upscattering were the dominant
source of $\gamma$-rays. This argues for a significant contribution from
$\gamma$-rays produced by the synchrotron-self-Compton process. We have found
that such strong contribution is possible if the jet is strongly clumped. We
reproduce the observed hard-state average broad-band spectrum using a
self-consistent jet model, taking into account all the relevant emission
processes, e$^\pm$ pair absorption, and clumping. This model also reproduces
the amplitude of the observed orbital modulation.
