We report the detection of CO(3-2) emission from a bright, gravitationally lensed Lyman Break Galaxy, LBGJ213512.73-010143 (the 'Cosmic Eye'), at z=3.07 using the Plateau de Bure Interferometer. This is only the second detection of molecular gas emission from an LBG and yields an intrinsic molecular gas mass of (2.4+/-0.4)x10^9 Mo. The lens reconstruction of the UV morphology of the LBG indicates that it comprises two components separated by ~2 kpc. The CO emission is unresolved, and appears to be centered on the intrinsically fainter (and also less highly magnified) of the two UV components. The width of the CO line indicates a dynamical mass of (8+/-2)x10^9csc(i)^2 Mo within the central 2 kpc. Employing mid-infrared observations from Spitzer we derive a stellar mass of ~(6+/-2)x10^9 Mo and a star-formation rate of ~60 Mo/yr, indicating that the molecular gas will be consumed in ~40 Myr. The gas fractions, star-formation efficiencies and line widths suggests that LBGJ213512 is a high-redshift, gas-rich analog of a local luminous infrared galaxy. This galaxy has a similar gas-to-dynamical mass fraction as observed in the submillimeter-selected population, although the gas surface density and star-formation efficiency is a factor of 3x less, suggesting less vigorous activity. We discuss the uncertainties in our conclusions arising from adopting a CO-to-H2 conversion factor appropriate for either the Milky Way or local luminous infrared galaxies. These observations demonstrate that current facilities, when aided by fortuitous gravitational magnification, can study 'ordinary' galaxies at high-redshift and so act as pathfinders for ALMA.