The Lagash I period is also the Early Dynastic IIIb period. The Lagash I dynasty ended with the reign of Uruinimgina (reigned 2351-2342 B.C.E.), who was defeated by Lugalzagesi of Uruk, who then controlled all of Mesopotamia (he also won Umma).
However, Lagash flourished again, particularly during the reign of Gudea (reigned c. 2141-2122 B.C.E.).