Is any dual metrizable locally convex space a Frechet space?

Question

The title basically says all of it.

If a normed space $F$ is a dual of a normed space $E$, then $F$ is a Banach space. I wonder if the same holds for Frechet spaces.

The strong dual $F$ of a locally convex space $E$ is complete, once $E$ is bornological, but I am not sure if this is the case here. Perhaps the completion of $E$ is though.