We identify SDSS J153636.22+044127.0, a QSO discovered in the Sloan Digital Sky Survey, as a promising candidate for a binary black hole system. This QSO has two broad-line emission systems separated by 3500 km/sec. The redder system at z=0.3889 also has a typical set of narrow forbidden lines. The bluer system (z=0.3727) shows only broad Balmer lines and UV Fe II emission, making it highly unusual in its lack of narrow lines. A third system, which includes only unresolved absorption lines, is seen at a redshift, z=0.3878, intermediate between the two emission-line systems. While the observational signatures of binary nuclear black holes remain unclear, J1536+0441 is unique among all QSOs known in having two broad-line regions, indicative of two separate black holes presently accreting gas. The interpretation of this as a bound binary system of two black holes having masses of 10^8.9 and 10^7.3 solar masses, yields a separation of ~ 0.1 parsec and an orbital period of ~100 years. The separation implies that the two black holes are orbiting within a single narrow-line region, consistent with the characteristics of the spectrum. This object was identified as an extreme outlier of a Karhunen-Loeve Transform of 17,500 z < 0.7 QSO spectra from the SDSS. The probability of the spectrum resulting from a chance superposition of two QSOs with similar redshifts is estimated at 2X10^-7, leading to the expectation of 0.003 such objects in the sample studied; however, even in this case, the spectrum of the lower redshift QSO remains highly unusual.
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