The angular distributions of the annihilation of photons were determined along the <111>, <110>, and <100> crystallographic directions in dislocation-free silicon grown by the floating zone method and subjected to annealing at 1050 and 1250-degrees-C in an atmosphere containing chlorine. Formation of positron-sensitive defects was observed. The results obtained were used to put forward a hypothesis on the formation of vacancy-type defects which could contain oxygen atoms. The recorded angular distributions could be regarded as a "direct" proof of the existence of a vacancy-enriched surface layer of silicon formed in the course of its heat treatment in an atmosphere containing chlorine.