Combine the first 4 ingredients. Rub the rosemary mixture over duck. Cover and chill 2 hours. Rinse duck with cold water; pat dry.
Heat oil in a large nonstick skillet over medium-high heat. Add duck, and cook for 5 minutes on each side or until done. Remove from pan; let stand for 10 minutes.
Combine the granulated sugar and vinegar in a small saucepan, and bring to a boil. Cook until thick and amber-colored (about 5 minutes). Add apricots; reduce heat, and cook for 1 minute or until the apricots begin to soften. Cut duck diagonally across the grain into slices. Serve with caramelized apricots.