Connelly was employed by
E. I. du Pont de Nemours and Company for 36 years. He joined the company in 1977 as a research engineer at the
DuPont Experimental Station in Wilmington, Delaware. He had assignments in Kentucky and West Virginia before starting his overseas assignments. He had positions in England, Switzerland and China – the final position with responsibility for DuPont's Asia Pacific businesses. He then returned to Wilmington in 1999 and was named vice president and general manager of DuPont Fluoroproducts. He was named senior vice-president of research and Chief Science and Technology Officer in 2001. He was promoted to Executive Vice President, the
Chief Innovation Officer and a member of the Office of the Chief Executives of DuPont in 2006. In this position, he had responsibility for DuPont's Applied BioSciences, Nutrition & Health, Performance Polymers and Packaging & Industrial Polymers businesses. He also had responsibility for Integrated Operations which includes Operations, Sourcing & Logistics and Engineering. DuPont announced he was retiring from the company in 2014. ==Other positions and honors==