  We introduce a property of compact complex manifolds under which the
existence of balanced metric is stable by small deformations of the complex
structure. This property, which is weaker than the
$\partial\overline\partial$-Lemma, is characterized in terms of the strongly
Gauduchon cone and of the first $\partial\overline\partial$-degree measuring
the difference of Aeppli and Bott-Chern cohomologies with respect to the Betti
number $b_1$.
