We present ALMA observations of the FW Tau system, a close binary pair of M5 stars with a wide-orbit (300 AU projected separation) substellar companion. The companion is extremely faint and red in the optical and near-infrared, but boasts a weak far-infrared excess and optical/near-infrared emission lines indicative of a primordial accretion disk of gas and dust. The component-resolved 1.3 mm continuum emission is found to be associated only with the companion, with a flux (1.78 +/- 0.03 mJy) that indicates a dust mass of 1-2 M_Earth. While this mass reservoir is insufficient to form a giant planet, it is more than sufficient to produce an analog of the Kepler-42 exoplanetary system or the Galilean satellites. The mass and geometry of the disk-bearing FW Tau companion remains unclear. Near-infrared spectroscopy shows deep water bands that indicate a spectral type later than M5, but substantial veiling prevents a more accurate determination of the effective temperature (and hence mass). Both a disk-bearing planetary-mass companion seen in direct light or a brown dwarf tertiary viewed in light scattered by an edge-on disk or envelope remain possibilities.