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 ≥60MeV of a previous work. The detection significance is 8σ in the 0.1-10 GeV range. In the soft state, we have found only upper limits on the emission at energies 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 γ-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 γ-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 γ-rays. This argues for a significant contribution from γ-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 hardstate 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.