We study the analogue in orbit equivalence of free product
decomposition and free indecomposability for countable groups. We
introduce the (orbit equivalence invariant) notion of freely
indecomposable (FI) standard probability measure preserving equivalence
relations and establish a criterion to check it, namely
non-hyperfiniteness and vanishing of the first L2-Betti
number. We obtain Bass-Serre rigidity results, i.e. forms of
uniqueness in free product decompositions of equivalence relations with
(FI) components. The main features of our work are weak algebraic
assumptions and no ergodicity hypothesis for the components. We deduce,
for instance, that a measure equivalence between two free products of
non-amenable groups with vanishing first l2-Betti numbers is
induced by measure equivalences of the components. We also deduce new
classification results in Orbit Equivalence and II_1 factors.