We present observations of Epsilon Eridani from the Submillimeter Array (SMA) at 1.3 millimeters and from the Australia Telescope Compact Array (ATCA) at 7 millimeters that reach an angular resolution of ~4 (13 AU). These first millimeter interferometer observations of Epsilon Eridani, which hosts the closest debris disk to the Sun, reveal two distinct emission components: (1) the well-known outer dust belt, which, although patchy, is clearly resolved in the radial direction, and (2) an unresolved source coincident with the position of the star. We use direct model-fitting of the millimeter visibilities to constrain the basic properties of these two components. A simple Gaussian shape for the outer belt fit to the SMA data results in a radial location of $64.4^{+2.4}_{-3.0}$ AU and FWHM of $20.2^{+6.0}_{-8.2}$ AU (fractional width $Delta R/R = 0.3$. Similar results are obtained taking a power law radial emission profile for the belt, though the power law index cannot be usefully constrained. Within the noise obtained (0.2 mJy/beam), these data are consistent with an axisymmetric belt model and show no significant azimuthal structure that might be introduced by unseen planets in the system. These data also limit any stellocentric offset of the belt to $<9$ AU, which disfavors the presence of giant planets on highly eccentric ($>0.1$) and wide (10s of AU) orbits. The flux density of the unresolved central component exceeds predictions for the stellar photosphere at these long wavelengths, by a marginally significant amount at 1.3 millimeters but by a factor of a few at 7 millimeters (with brightness temperature $13000 pm 1600$ K for a source size of the optical stellar radius). We attribute this excess emission to ionized plasma from a stellar corona or chromosphere.
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