A new method of determining the thickness of mare basalts on the Moon is introduced that is made possible by high-resolution gravity data acquired from NASA's Gravity Recovery and Interior Laboratory (GRAIL) mission. Using a localized multitaper spherical-harmonic analysis, an effective density spectrum is calculated that provides an estimate of the average crustal density as a function of spherical harmonic degree. By comparing the observed effective density spectrum with one generated from a theoretical model, the thickness of mare basalts can be constrained. We assume that the grain density of the basalts is known from remote sensing data and petrologic considerations, we assign a constant porosity to the basalts, and we let both the thickness of the basalts and the density of the underlying crust vary. Using this method, the total thickness of basalts was estimated on the nearside hemisphere, yielding an average of 0.74 km with 1σ upper and lower bounds of 1.62 km and 100 m, respectively. The region of Marius Hills, which is a long-lived volcanic complex, is found to have the thickest basalts, with an average of 2.86 km and 1σ limits of 3.65 and 1.02 km, respectively. The crust beneath the Mare Imbrium basalts is found to have an atypically high density of about 3000 kg m-3 that we interpret as representing a mafic, unfractured, impact melt sheet.