Quality Assurance in the Private Sector Drug Testing Act of 1993 - Amends the Public Health Service Act to prohibit a laboratory from performing a drug test in connection with any drug testing program subject to this Act unless the laboratory is certified under this Act. Directs the Secretary of Health and Human Services to establish standards for certifying laboratories. Requires an employer, as a condition of implementing or maintaining a drug testing program, to establish a written anti-drug abuse policy containing specified elements and a drug-free awareness program. Sets forth employee protections. Allows professional athletes to be treated as other employees under this Act except for certain provisions relating to confidentiality.