Starter and starting method of compression self-ignition engine

When a compression self-ignition engine is automatically stopped, at the same time as fuel cut for stopping fuel injection from a fuel injection valve (15) is executed, an operating position of an intake throttle (30) is set to fully closed, and the fully-closed state is maintained at least until final TDC. Furthermore, an operation of an accessory (32) performed after the fuel cut is controlled to adjust an engine load so that an engine rotation speed (Net) when passing the final TDC falls within a specific speed range (P) determined in advance. Accordingly, a piston (5) of a Compression-Stop Cylinder (2C) is stopped with high accuracy at a target position that is set on a bottom dead center-side of an intermediate position (CAx) between top dead center and bottom dead center, and a restart time of the engine is thereby reduced.