We describe a directed search for continuous gravitational waves in data from the sixth initial LIGO science run. The target was the nearby globular cluster NGC 6544 at a distance of $\approx 2.7\mathrm{kpc}$. The search covered a broad band of frequencies along with first and second frequency derivatives for a fixed sky position. The search coherently integrated data from the two LIGO interferometers over a time span of 9.2 days using the matched-filtering $F$-statistic. We found no gravitational-wave signals and set 95% confidence upper limits as stringent as $6.0\times {10}^{-25}$ on intrinsic strain and $8.5\times {10}^{-6}$ on fiducial ellipticity. These values beat the indirect limits from energy conservation for stars with characteristic spin-down ages older than 300 years and are within the range of theoretical predictions for possible neutron-star ellipticities. An important feature of this search was use of a barycentric resampling algorithm which substantially reduced computational cost; this method is used extensively in searches of Advanced LIGO and Virgo detector data