We present the discovery of an extremely-luminous dust-obscured galaxy (DOG) at $z_{rm spec}$ = 3.703, WISE J101326.25+611220.1. This DOG is selected as a candidate of extremely-luminous infrared (IR) galaxies based on the photometry from the Sloan Digital Sky Survey and Wide-field Infrared Survey Explorer. In order to derive its accurate IR luminosity, we perform follow-up observations at 450 and 850 $mu$m using the Submillimetre Common User Bolometer Array 2 on the James Clerk Maxwell Telescope, and at 870 and 1300 $mu$m using the Submillimeter Array, which enable us to pin down its IR Spectral Energy Distribution (SED). We perform SED fitting using 14 photometric data (0.4 - 1300 $mu$m) and estimate its IR luminosity, $L_{rm IR}$ (8-1000 $mu$m), to be $2.2^{+1.5}_{-1.0}$ $times 10^{14}$ $L_{odot}$, making it one of the most luminous IR galaxies in the Universe. The energy contribution from an active galactic nucleus (AGN) to the IR luminosity is $94^{+6}_{-20}$%, which indicates it is an AGN-dominated DOG. On the other hand, its stellar mass ($M_*$) and star formation rate (SFR) are $log ,(M_ast/M_{odot})$ = $11.2^{+0.6}_{-0.2}$ and $log ,({rm SFR}/M_{odot},{rm yr}^{-1}$) = $3.1^{+0.2}_{-0.1}$, respectively, which means that this DOG can be considered as a starburst galaxy in $M_*$--SFR plane. This extremely-luminous DOG shows significant AGN and star forming activity that provides us an important laboratory to probe the maximum phase of the co-evolution of galaxies and supermassive black holes.
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