Layer thickness measurements with an acoustic micrometer using pseudo-Sezawa waves in which ultrasonic waves are obliquely applied to a layered surface of a specimen have been proposed. A case in which the plate thickness of the specimen is so thin that it cannot be regarded as a half space is studied. A number of modes of plate waves are then excited in addition to pseudo-Sezawa waves. The plate waves, giving rise to the appearance of extra dips in the power spectrum of reflected waves, cause difficulties in the measurements. To prevent the excitation of plate waves, it is proposed that a mask of a sound-insulating material with a slit aperture should be placed on the layered surface of the specimen. Experiments and theoretical calculations, using lead frames of LSI chips as typical test specimens with thin substrates, were performed to demonstrate the effectiveness of the present method in preventing the excitation of plate waves.