Authors:Cleeves et alAbstract:The spatial distribution of gas and solids in protoplanetary disks determines the composition and formation efficiency of planetary systems. A number of disks show starkly different distributions for the gas and small grains compared to millimeter-centimeter sized dust. We present new Atacama Large Millimeter/Submillimeter Array (ALMA) observations of the dust continuum, CO, 13CO, and C18O in the IM Lup protoplanetary disk, one of the first systems where this dust-gas dichotomy was clearly seen. The 12CO is detected out to a radius of 970 AU, while the millimeter continuum emission is truncated at just 313 AU. Based upon this data, we have built a comprehensive physical and chemical model for the disk structure, which takes into account the complex, coupled nature of the gas and dust and the interplay between the local and external environment. We constrain the distributions of gas and dust, the gas temperatures, the CO abundances, the CO optical depths, and the incident external radiation field. We find that the reduction/removal of dust from the outer disk exposes this region to higher stellar and external radiation and decreases the rate of freeze-out, allowing CO to remain in the gas out to large radial distances. We estimate a gas-phase CO abundance of 5% of the ISM value and a low external radiation field (G0≲4). The latter is consistent with that expected from the local stellar population. We additionally find tentative evidence for ring-like continuum substructure, suggestions of isotope-selective photodissociation, and a diffuse gas halo.