We present new mass models for the gravitational lens system B1938+666, using multiwavelength data acquired from Keck adaptive optics (AO) and Hubble Space Telescope (HST) observations. These models are the first results from the Strong lensing at High Angular Resolution Program (SHARP), a project designed to study known quadruple image and Einstein ring lenses using high-resolution imaging, in order to probe their mass distributions in unprecedented detail. Here, we specifically highlight differences between AO- and HST-derived lens models, finding that – at least when the lens and source galaxies are both bright and red, and the system has a high degree of circular symmetry – AO-derived models place significantly tighter constraints on model parameters. Using this improved precision, we infer important physical properties about the B1938+666 system, including the mass density slope of the lensing galaxy (γ = 2.045), the projected dark matter mass fraction within the Einstein radius (Mdark/Mlens = 0.55) and the total magnification factor of the source galaxy (∼13). Additionally, we measure an upper-limit constraint on luminous substructure (MV > 16.2), based on the non-detection of bright satellite galaxies in all data sets. Finally, we utilize the improved image resolution of the AO data to reveal the presence of faint arcs outside of the primary Einstein ring. The positions and orientations of these arcs raise the intriguing possibility that B1938+666 has a second source galaxy, located at a more distant redshift. However, future work is needed to verify this hypothesis.