A type III${}_1$ factor with the smallest outer automorphism group

The canonical modular homomorphism provides an embedding of $\mathbb{R}$ into the outer automorphism group Out($M$) of any type III${}_1$ factor $M$. We provide an explicit construction of a full factor of type III${}_1$ with separable predual such that the outer automorphism group is minimal, i.e. this embedding is an isomorphism. We obtain such a III${}_1$ factor by using an amalgamated free product construction.