We show that the perturbative g invariant of rational homology 3-spheres can be recovered from the Le-Murakami-Ohtsuki (LMO) invariant for any simple Lie algebra g, that is, the LMO invariant is universal among the perturbative invariants. This universality was conjectured in Le, Murakami and Ohtsuki ['On a universal perturbative invariant of 3-manifolds', Topology 37 (1998) 539-574]. Since the perturbative invariants dominate the quantum invariants of integral homology 3-spheres [K. Habiro, 'On the quantum sl(2) invariants of knots and integral homology spheres', Invariants of knots and 3-manifolds (Kyoto 2001), Geometry and Topology Monographs 4 (Geometry and Topology Publications, Coventry, 2002) 161-181; K. Habiro, 'A unified Witten-Reshetikhin-Turaev invariant for integral homology spheres', 171 (2008) 1-81; K. Habiro and T. T. Q. Le, in preparation] the LMO invariant dominates the quantum invariants of integral homology 3-spheres.