Graphene, a single monolayer of graphite, has recentlyattracted considerable interest owing to its novel magneto-transport properties(1-3), high carrier mobility and ballistic transport up to room temperature(4). It has the potential for technological applications as a successor of silicon in the post Moore's law era(5-7), as a single-molecule gas sensor(8), in spintronics(9-11), in quantum computing(12) or as a terahertz oscillator(13). For such applications, uniform ordered growth of graphene on an insulating substrate is necessary. The growth of graphene on insulating silicon carbide (SiC) surfaces by high-temperature annealing in vacuum was previously proposed to open a route for large-scale production of graphene-based devices(5,6). However, vacuum decomposition of SiC yields graphene layers with small grains (30-200 nm; refs 14-16). Here, we show that the ex situ graphitization of Si-terminated SiC(0001) in an argon atmosphere of about 1 bar produces monolayer graphene films with much larger domain sizes than previously attainable. Raman spectroscopy and Hall measurements confirm the improved quality of the films thus obtained. High electronic mobilities were found, which reach mu = 2,000 cm(2) V(-1) s(-1) at T = 27 K. The new growth process introduced here establishes a method for the synthesis of graphene films on a technologically viable basis.