  We present the first results of an all-sky search for continuous
gravitational waves from unknown spinning neutron stars in binary systems using
LIGO and Virgo data. Using a specially developed analysis program, the TwoSpect
algorithm, the search was carried out on data from the sixth LIGO Science Run
and the second and third Virgo Science Runs. The search covers a range of
frequencies from 20 Hz to 520 Hz, a range of orbital periods from 2 to ~2,254 h
and a frequency- and period-dependent range of frequency modulation depths from
0.277 to 100 mHz. This corresponds to a range of projected semi-major axes of
the orbit from ~0.6e-3 ls to ~6,500 ls assuming the orbit of the binary is
circular. While no plausible candidate gravitational wave events survive the
pipeline, upper limits are set on the analyzed data. The most sensitive 95%
confidence upper limit obtained on gravitational wave strain is 2.3e-24 at 217
Hz, assuming the source waves are circularly polarized. Although this search
has been optimized for circular binary orbits, the upper limits obtained remain
valid for orbital eccentricities as large as 0.9. In addition, upper limits are
placed on continuous gravitational wave emission from the low-mass x-ray binary
Scorpius X-1 between 20 Hz and 57.25 Hz.
