We perform a search for gravitational wave bursts using data from the second science run of the LIGO detectors, using a method based on a wavelet time-frequency decomposition. This search is sensitive to bursts of duration much less than a second and with frequency content in the 100-1100 Hz range. It features significant improvements in the instrument sensitivity and in the analysis pipeline with respect to the burst search previously reported by LIGO. Improvements in the search method allow exploring weaker signals, relative to the detector noise floor, while maintaining a low false alarm rate, O(0.1) μHz. The sensitivity in terms of the root-sum-square (rss) strain amplitude lies in the range of hrss˜10-20-10-19Hz-1/2. No gravitational wave signals were detected in 9.98 days of analyzed data. We interpret the search result in terms of a frequentist upper limit on the rate of detectable gravitational wave bursts at the level of 0.26 events per day at 90% confidence level. We combine this limit with measurements of the detection efficiency for selected waveform morphologies in order to yield rate versus strength exclusion curves as well as to establish order-of-magnitude distance sensitivity to certain modeled astrophysical sources. Both the rate upper limit and its applicability to signal strengths improve our previously reported limits and reflect the most sensitive broad-band search for untriggered and unmodeled gravitational wave bursts to date.