I'm not sure if this is research level, so feel free to vote to migrate.

Suppose we have a complete boolean algebra $A$, with a _dense_, $\sigma$-complete subalgebra $B$, and a $\sigma$-complete homomorphism $h : B \to C$, where $C$ is complete.  Does $h$ have a $\sigma$-complete extension $h' : A \to C$?

EDIT: I forgot to say that I'm interested in the case that $A$ is atomless.