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 \sim 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 \sim 0.6 \times 10 ^ { -3 } ls to \sim 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 2.3 \times 10 ^ { -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 .