Due to its thermodynamic reversibility, the Stirling cycle has an inherent advantage in thermal efficiency over the standard, irreversible standing-wave thermoacoustic cycle. The main disadvantage of the Stirling cycle has been the need for high-pressure, sliding piston seals. Building significantly on the ideas of Ceperley [J. Acoust. Soc. Am. 66, 1508 (1979)], a thermoacoustic version of a 1-kW Stirling engine has been constructed and tested. This engine enjoys both the efficiency advantage of the Stirling cycle and the no-moving-parts advantage of thermoacoustics. In this first attempt, experimental data show that the engine has an efficiency above 35% of the Carnot efficiency, much higher than that of any prior thermoacoustic engine. [Work supported by the Office of Basic Energy Sciences in the U.S. DOE.]