  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 $L^2$-Betti
number. We obtain Bass-Serre rigidity results, \textit{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 $\ell^2$-Betti numbers is induced by measure equivalences of
the components. We also deduce new classification results in Orbit Equivalence
and II$_1$ factors.
