Whether a Brahmana is determined by birth?

Apastamba dharma sUtra clearly establishes that the VarNa is determined by birth. Accordingly a BrAhmaNa is also determined by birth.

catvAro varNo brAhmaNa kshatriya vaiSya SUdrAh ! i-3
tesham pUrvah pUrvo janmatah SreyAn ! i-4