By measuring AC susceptibility using a very low amplitude of the AC field (<1 mG) it is shown that underdoped samples of La2-chiSrchiCuO4, are diamagnetic in a temperature region above T-c up to a temperature T* when the field is directed along the c-axis. At T* a broad step anomaly in the specific heat is inferred through measurements of the elastic constant c(33). We suggest that the observed diamagnetism and the anomaly in the elastic constant are associated with the existence of phase incoherent Cooper pairs between T-c and T*. (C) 2004 Elsevier B.V. All rights reserved.