Finding a vector potential, i.e. given $\vec{A}$, how to find $\vec{B}$ s.t. $\vec{A} = \nabla \times \vec{B}$?

Question

I understand that this might not be unique, but is there a (relatively) painless way to generate such a 'vector potential', so for a given field $\vec{A}$, a new field $\vec{B}$ which satifies:

$$\vec{A} = \nabla \times \vec{B}$$