In 1998, Connelly ran for Maine's first district House of Representatives seat on the Republican ticket and lost to fellow Republican James Longley. In 2000, Connelly directed the Maine branch of George W. Bush's presidential campaign. In October 2001, Bush rewarded Connelly by appointing him to serve on the board of OPIC, the government office that provides political insurance to U.S. companies who want to invest in unstable parts of the world.
Since his appointment to the OPIC board, Connelly has traveled to Afghanistan, Russia, Azerbaijan, Serbia, Peru, and the Middle East to promote U.S. overseas investments, in particular the investments of U.S. companies (including Bechtel) in the privatization of foreign oil, power, water, and transportation infrastructures.
Connelly has attracted criticism for his role in OPIC projects and investments which critics say will directly benefit his former employers. In his tenure at OPIC, Connelly finalized OPIC's $130 million loan to Lima Airport Partners, a joint venture between Bechtel to privatize Jorge Chávez International Airport in Lima. He has also been involved in OPIC's investments in Serbia, a country where Bechtel has been awarded large highway building contracts.
Connelly earned his Bachelor's degree in 1974 from Duke University and his Master's degree from Georgetown University's School of Foreign Service in 1977. He and his family have lived near Biddeford, Maine for over 30 years.