The answer to the question, as asked, is "no".  For, suppose that $B$ is a three-ball. So, $B$ is a genus zero handlebody.  Let $\alpha$ be a knotted arc properly embedded in $B$.  So the fundamental group of $B - \alpha$ is not free.  Let $A$ be the frontier of a regular neighbourhood of $\alpha$.  So $\alpha$ is an annulus, properly embedded in $B$.  Cutting $B$ along $A$ and taking closures yields a three-ball ("inside $A$") and a knot complement ("outside $A$").