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.