  The renormalization of a gapless Phi-derivable Hartree--Fock approximation to
the O(N)-symmetric lambda*phi^4 theory is considered in the spontaneously
broken phase. This kind of approach was proposed by three of us in a previous
paper in order to preserve all the desirable features of Phi-derivable
Dyson-Schwinger resummation schemes (i.e., validity of conservation laws and
thermodynamic consistency) while simultaneously restoring the Nambu--Goldstone
theorem in the broken phase. It is shown that unlike for the conventional
Hartree--Fock approximation this approach allows for a scale-independent
renormalization in the vacuum. However, the scale dependence still persists at
finite temperatures. Various branches of the solution are studied. The
occurrence of a limiting temperature inherent in the renormalized Hartree--Fock
approximation at fixed renormalization scale mu is discussed.
