The boundary with the North Riding of Yorkshire was adjusted : the part of the town of Barnard Castle that was in Yorkshire was added to County Durham , but the part of the Borough of Stockton-on-Tees in Durham was given to the North Riding . The county boundary with the North Riding of Yorkshire was adjusted : that part of the town of Barnard Castle historically in Yorkshire was added to County Durham , while the portion of the Borough of Stockton-on-Tees in Durham was ceded to the North Riding .