Stopping Trained in America Ph.D.s From Leaving the Economy Act of 2013 or the STAPLE Act - Amends the Immigration and Nationality Act to exempt from: (1) direct numerical limitations aliens who have earned a Ph.D. degree in science, technology, engineering, or mathematics (STEM) from a U.S. institution of higher education and who have an offer of employment from a U.S. employer in a field related to such degree; and (2) H-1B visa (specialty occupation) numerical limitations aliens who have a U.S. STEM Ph.D. and with respect to whom the petitioning employer requires such an education.