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/Spectral and Photometric Imaging Receiver (SPIRE) and APEX/LABOCA data as an extremely red "870 mu m riser" (i.e., S-250 (mu m)< S-350 mu m< S-500 mu m< S-870 mu m), demonstrating the utility of this technique to identify some of the highest-redshiftdusty galaxies. A scan of the 3 mm atmospheric window with ALMA yields detections of CO(J = 5 -> 4) and CO(J = 6 -> 5) emission, and a tentative detection of H2O(2(11)-> 2(02)) emission, which provides anunambiguous redshift measurement. The strength of the CO lines implies a large molecular gas reservoir with a mass of M-gas = 2.5 x 10(11) (alpha(CO)/0.8)(0.39/(r51)) M-circle dot sufficient to maintain its similar to 2400Me yr(-1)starburst for at least similar to 100 Myr. The 870 mu m 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 similar or equal to 2.4 x. 10(13) L-circle dot implies that this system represents a binary hyper-luminous infrared galaxy, themost distant of its kind presently known. This also implies star formation rate surface densities of Sigma(SFR) = 730 and 750Me yr(-1) kpc(2), 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 similar to 6 quasars and massive quiescent galaxies at z greater than or similar to 3.