The Very Large Array was used to observe the ultracool rapidly rotating M9 dwarf TVLM 513-46546 simultaneously at 4.88 and 8.44 GHz. The radio emission was determined to be persistent, variable, and periodic at both frequencies with a period of ~2 hr. This periodicity is in excellent agreement with the estimated period of rotation of the dwarf based on its v sin i of ~60 km s^-1. This rotational modulation places strong constraints on the source size of the radio-emitting region and hence the brightness temperature of the associated emission. We find the resulting high brightness temperature, together with the inherent directivity of the rotationally modulated component of the emission, difficult to reconcile with incoherent gyrosynchrotron radiation. We conclude that a more likely source is coherent, electron cyclotron maser emission from the low-density regions above the magnetic poles. This model requires the magnetic field of TVLM 513-46546 to take the form of a large-scale, stable dipole or multipole with surface field strengths up to at least 3 kG. We discuss a mechanism by which broadband, persistent electron cyclotron maser emission can be sustained in the low-density regions of the magnetospheres of ultracool dwarfs. A second nonvarying, unpolarized component of the emission may be due to depolarization of the coherent electron cyclotron maser emission or, alternatively, incoherent gyrosynchrotron or synchrotron radiation from a population of electrons trapped in the large-scale magnetic field.