We demonstrate a chaotic microcavity laser whose quality is comparable to typical nonchaotic microcavity lasers, yet it has the unique characteristic of unidirectional output. The cavity shape is a disk with the boundary defined by a curve called limaccedilon of Pascal. For a lasing mode of volume less than 0.2 mu m(3), the measured quality factor is 23 000. The record-high quality and small modal volume result in extremely low-lasing threshold. The spontaneous emission coupling efficiency is approximately 6% and is the reported value for a chaotic microcavity laser. All the lasing modes have one output beam in the same direction with a divergence angle less than 40 degrees. This universal directionality is determined by the chaotic ray dynamics in the open cavity.