We have obtained a firm detection of Cyg X-1 during its hard and intermediate spectral states in the energy range of 40MeV-60 GeV based on observations by the Fermi Large Area Telescope, confirming the independent results at >= 60MeV of a previous work. The detection significance is similar or equal to 8 sigma in the 0.1-10 GeV range. In the soft state, we have found only upper limits on the emission at energies greater than or similar to 0.1MeV. However, we have found emission with a very soft spectrum in the 40-80MeV range, not detected previously. This is likely to represent the high-energy cut-off 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(+/-) pair absorption and clumping. This model also reproduces the amplitude of the observed orbital modulation.