Complex sentence: Part of the county--the so-called "Barrois mouvant"--became a fief of the Kingdom of France in 1301, while the "Barrois non-mouvant" remained a part of the Empire.
Split into simpler sentences: Part of the county, the so-called "Barrois mouvant", became a fief of the Kingdom of France in 1301. The "Barrois non-mouvant" remained a part of the Empire.