It is argued that the Schwarzschild black hole solution follows as a u
nique limit of the Brans-Dicke Class I solutions, provided the correct
iterated limit is taken. Such a uniqueness is essential for the valid
ity of a recent version of the no-hair conjecture. A non-trivial modif
ication to this version is proposed in order to exclude Brans-Dicke Cl
ass IV solutions which appear to represent scalar hair black holes in
general.