Let $H_1$ and $H_2$ be Hilbert spaces.

Let $A\subset B(H_1)$ be a factor and $A'$ its commutant.

If a von Neumann algebra $M\subset B(H_1\otimes H_2)$ contains $A\otimes 1$ and commutes with $A'\otimes 1$, is it then necessarily of the form $M=A\otimes B$ for some von Neumann algebra $B\subset B(H_2)$?

I think that I know how to prove this if $A$ is hyperfinite, and I wonder if it's true in general.