We present the results obtained from an all-sky search for gravitational-wave (GW) bursts in the 64-2000 Hz frequency range in data collected by the LIGO detectors during the first year (November 2005-November 2006) of their fifth science run. The total analyzed live time was 268.6 days. Multiple hierarchical data analysis methods were invoked in this search. The overall sensitivity expressed in terms of the root-sum-square (rss) strain amplitude hrss for gravitational-wave bursts with various morphologies was in the range of 6×10-22 Hz-1/2 to a few×10-21 Hz-1/2. No GW signals were observed and a frequentist upper limit of 3.75 events per year on the rate of strong GW bursts was placed at the 90% confidence level. As in our previous searches, we also combined this rate limit with the detection efficiency for selected waveform morphologies to obtain event rate versus strength exclusion curves. In sensitivity, these exclusion curves are the most stringent to date.