After high school, Leddy took a short business course in Miami, Florida, and then, in 1933, moved to Washington, D.C., to live with an aunt.

During this time, Leddy worked at the Home Owners' Loan Corporation and later the Pan American Union during the day, and took classes at Georgetown University at night.

He then worked with State Department lawyer Marc Catudal, an expert on the most favored nation clause, on the legal framework for trade agreements.

In 1961, President of the United States John F. Kennedy named Leddy as Assistant Secretary of the Treasury for International Affairs.

President Lyndon B. Johnson nominated him as Assistant Secretary of State for European Affairs in 1965, with Leddy holding this office from June 16, 1965, until February 19, 1969.