  We report the detection of ADFS-27, a dusty, starbursting major merger at a
redshift of z=5.655, using the Atacama Large Millimeter/submillimeter Array
(ALMA). ADFS-27 was selected from Herschel/SPIRE and APEX/LABOCA data as an
extremely red "870 micron riser" (i.e., S_250<S_350<S_500<S_870), demonstrating
the utility of this technique to identify some of the highest-redshift dusty
galaxies. A scan of the 3mm atmospheric window with ALMA yields detections of
CO(5-4) and CO(6-5) emission, and a tentative detection of H2O(211-202)
emission, which provides an unambiguous redshift measurement. The strength of
the CO lines implies a large molecular gas reservoir with a mass of
M_gas=2.5x10^11(alpha_CO/0.8)(0.39/r_51) Msun, sufficient to maintain its ~2400
Msun/yr starburst for at least ~100 Myr. The 870 micron dust continuum emission
is resolved into two components, 1.8 and 2.1 kpc in diameter, separated by 9.0
kpc, with comparable dust luminosities, suggesting an ongoing major merger. The
infrared luminosity of L_IR~=2.4x10^13Lsun implies that this system represents
a binary hyper-luminous infrared galaxy, the most distant of its kind presently
known. This also implies star formation rate surface densities of Sigma_SFR=730
and 750Msun/yr/kpc2, consistent with a binary "maximum starburst". The
discovery of this rare system is consistent with a significantly higher space
density than previously thought for the most luminous dusty starbursts within
the first billion years of cosmic time, easing tensions regarding the space
densities of z~6 quasars and massive quiescent galaxies at z>~3.
