Are these two definitions of closable linear operators equivalent?

Question

I have come across two different definitions for closable linear operators.

1. A densely defined linear operator $A$ is called closable if its adjoint $A^\ast$ is also densely defined.
2. Suppose that there exists a closed linear operator $B$ that extends a linear operator $A:D(A) \to H$. Then $A$ is closable.

Can an 'if and only if' relationship be shown between 1. and 2. or there some differences in these definitions?