LS I +61 303 is one of only a few high-mass X-ray binaries currently detected at high significance in very high energy γ-rays. The system was observed over several orbital cycles (between 2006 September and 2007 February) with the VERITAS array of imaging air Cerenkov telescopes. A signal of γ-rays with energies above 300 GeV is found with a statistical significance of 8.4 standard deviations. The detected flux is measured to be strongly variable; the maximum flux is found during most orbital cycles at apastron. The energy spectrum for the period of maximum emission can be characterized by a power law with a photon index of Γ = 2.40 ± 0:16stat ± 0.2sys and a flux above 300 GeV corresponding to 15%–20% of the flux from the Crab Nebula.