  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.
