We have produced ultrathin epitaxial graphite films which show remarkable 2D electron gas (2DEG) behavior. The films, composed of typically three graphene sheets, were grown by thermal decomposition on the (0001) surface of 6H-SiC, and characterized by surface science techniques. The low-temperature conductance spans a range of localization regimes according to the structural state (square resistance 1.5 kOmega to 225 kOmega at 4 K, with positive magnetoconductance). Low-resistance samples show characteristics of weak localization in two dimensions, from which we estimate elastic and inelastic mean free paths. At low field, the Hall resistance is linear up to 4.5 T, which is well-explained by n-type carriers of density 10(12) cm(-2) per graphene sheet. The most highly ordered sample exhibits Shubnikov-de Haas oscillations that correspond to nonlinearities observed in the Hall resistance, indicating a potential new quantum Hall system. We show that the high-mobility films can be patterned via conventional lithographic techniques, and we demonstrate modulation of the film conductance using a top-gate electrode. These key elements suggest electronic device applications based on nanopatterned epitaxial graphene (NPEG), with the potential for large-scale integration.