We present an all-sky search for gravitational waves in the frequency range 1 to 6 kHz during the first calendar year of LIGO’s fifth science run. This is the first untriggered LIGO burst analysis to be conducted above 3 kHz. We discuss the unique properties of interferometric data in this regime. 161.3 days of triple-coincident data were analyzed. No gravitational events above threshold were observed and a frequentistupper limit of 5.4 year^-1 on the rate of strong gravitational-wave bursts was placed at a 90% confidence level. Implications for specific theoretical models of gravitational-wave emission are also discussed.