We report the detection of CO(3-2) emission from a bright, gravitationally lensed Lyman Break galaxy, LBG J213512.73-010143 (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)×109 Msolar. 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, ?<~3'', 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)×109csc2i Msolar within the central 2 kpc. Employing mid-infrared observations from Spitzer, we infer a stellar mass of M*~(6+/-2)×109 Msolar and a star formation rate of ~60 Msolar yr-1, indicating that the molecular gas will be consumed in <~40 Myr. The gas fractions, star formation efficiencies, and line widths suggests that LBG J213512 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 3 times 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.