  In this paper we study asymptotic Gibbs measures in the diluted p-spin models
in the so called 1-RSB case, when the overlap takes two values $q_*, q^*\in
[0,1].$ When the external field is not present and the overlap is not equal to
zero, we prove that such asymptotic Gibbs measures are described by the
M\'ezard-Parisi ansatz conjectured in [MP]. When the external field is present,
we prove that the overlap can not be equal to zero and all 1-RSB asymptotic
Gibbs measures are described by the M\'ezard-Parisi ansatz. Finally, we give a
characterization of the exceptional case when there is no external field and
the smallest overlap value is equal to zero, although it does not go as far as
the M\'ezard-Parisi ansatz. Our approach is based on the cavity computations
combined with the hierarchical exchangeability of pure states.
