  We report a synchrotron x-ray scattering study of the diluted spin-Peierls
(SP) material Cu_(1-x)Mg_xGeO_3. We find that for x>0 the temperature T_m at
which the spin gap is established is significantly higher than the temperature
T_s at which the SP dimerization attains long-range order. The latter is
observed only for x<x_c =0.021, while for x>x_c the SP correlation length
quickly decreases with increasing x. We argue that impurity-induced competing
interactions play a central role in these phenomena.
