  We generalize the Hitchin-Kobayashi correspondence between semistability and
the existence of approximate Hermitian-Yang-Mills structures to the case of
principal Higgs bundles. We prove that a principal Higgs bundle on a compact
Kaehler manifold, with structure group a connected linear algebraic reductive
group, is semistable if and only if it admits an approximate
Hermitian-Yang-Mills structure.
