We report the results of a directed search for continuous gravitational-wave emission in a broad frequency range (between 50 and 1000 Hz) from the central compact object of the supernova remnant Cassiopeia A (Cas A). The data come from the sixth science run of LIGO, and the search is performed on the volunteer distributed computing network Einstein@Home. We find no significant signal candidate and set the most constraining upper limits to date on the gravitational-wave emission from Cas A, which beat the indirect age-based upper limit across the entire search range. At 170 Hz (the most sensitive frequency range), we set 90% confidence upper limits on the gravitational-wave amplitude h0 of ∼2.9×10−25, roughly twice as constraining as the upper limits from previous searches on Cas A. The upper limits can also be expressed as constraints on the ellipticity of Cas A; with a few reasonable assumptions, we show that at gravitational-wave frequencies greater than 300 Hz we can exclude an ellipticity of ≳10−5.