  We give a new proof of a theorem due to Alain Connes, that an injective
factor $N$ of type III$_1$ with separable predual and with trivial
bicentralizer is isomorphic to the Araki--Woods type III$_1$ factor
$R_{\infty}$. This, combined with the author's solution to the bicentralizer
problem for injective III$_1$ factors provides a new proof of the theorem that
up to $*$-isomorphism, there exists a unique injective factor of type III$_1$
on a separable Hilbert space.
