We describe a directed search for continuous gravitational waves in data from the sixth initial LIGOscience run. The target was the nearby globular cluster NGC 6544 at a distance of ≈2.7 kpc. The searchcovered a broad band of frequencies along with first and second frequency derivatives for a fixed skyposition. The search coherently integrated data from the two LIGO interferometers over a time span of9.2 days using the matched-filtering F-statistic. We found no gravitational-wave signals and set95% confidence upper limits as stringent as 6.0 × 10−25 on intrinsic strain and 8.5 × 10−6 on fiducialellipticity. These values beat the indirect limits from energy conservation for stars with characteristic spindownages older than 300 years and are within the range of theoretical predictions for possible neutron-starellipticities. An important feature of this search was use of a barycentric resampling algorithm whichsubstantially reduced computational cost; this method is used extensively in searches of Advanced LIGOand Virgo detector data.