Concealed Weapons Prohibition Act of 1996 - Prohibits a person from carrying a handgun on his or her person in public. Sets forth exceptions where a person is authorized to carry a handgun: (1) under Federal law; (2) under a State law that grants an exemption based on an individualized determination and a review of credible evidence that the person should be allowed to carry a handgun because of compelling circumstances; or (3) under a State law that provides an exemption for any person who is a law enforcement or retired law enforcement official, a duly authorized private security officer, a person whose employment involves the transport of substantial amounts of cash or other valuables, or anyone the Attorney General determines should be allowed to carry a handgun because of compelling circumstances warranting an exception, pursuant to regulations that the Attorney General may promulgate.