The extreme carbon star, AFGL 3068, is losing mass at a rate in excess of 10$^{-4}$ M$_{\odot}$ yr$^{-1}$, and has so far been detected only in the infrared because it is hidden by a thick dust photosphere having a color temperature of $\sim$300K. Using the ACS camera on HST, we have imaged AFGL 3068 with broad-band filters at 0.6 and 0.8 $\mu$m and find a thin, apparently continuous spiral arc winding 4 or 5 times around the location of the star, from angular radii of 2 to 10 arcsec. We interpret this as the projection of nested spiral shells such as were predicted to occur when the mass-losing star is a member of a binary system. In this case, the illumination is presumably provided by ambient galactic starlight. Subsequent near-IR observations with the NIRC2 camera on the Keck II telescope using adaptive optics reveal that AFGL 3068 has two components separated by 0.11 arcsec, or 109 AU at a distance of 1 kpc. One very red component is presumably the mass-losing carbon star, while the other component is apparently a much bluer companion. Assuming each component has mass M(M$_{\odot}$), and ignoring the projection of the separation vector, we find the binary period to be 810 M$^{-0.5}$ yrs, strikingly comparable to the 710-yr separation of the shells obtained from the known outflow velocity of 14.7 km s$^{-1}$.