The answer is that it depends on the domain of definition of the vector field. For vector fields defined on all of R³, it is true that there exists a vector field A such that B = curl A. If B is merely defined on R³\{0} (and cannot be extended smoothly to all of R³), there may not exist such a vector field.
You could also have asked, given that curlA=0, does there exist a function f such that A=grad(f) ?
The general setting for answering questions such as these concerning the existence of a "primitive" in some sense or another is the subject of the de Rham theory of differential forms. In this context, to ask whether every smooth vector field of vanishing divergence defined on some open subset U of R³ is equal to the curl of some other vector fields is the same as asking whether the second de Rham cohomology group of U is the trivial group: H^2_{\mbox{de Rham}}(U)=0. Now, it is known that the de Rham cohomology groups are homotopy invariants: two homotopy equivalent spaces have isomorphic de Rham cohomology groups. In particular, R³ is homotopy equivalent to a point, which have vanishing de Rham cohomology, and so H^2_{\mbox{de Rham}}(\mathbb{R}^3)=0, which translated into the fact that for vector fields defined on all of R³ of vanishing divergence, it is true that there exists a vector field A such that B = curl A. But for R³\{0}, there are known counter-examples. One of them translates in the following way in the terms that interests you.
Let B:R³\{0}-->R³ be the vector field
B(x,y,z)=\frac{x\hat{x}+y\hat{y}+z\hat{z}}{(x^2+y^2+z^2)^{3/2}}=\frac{r}{|r|^3}
Suppose that B=curl(A) for some vector field A:R³\{0}-->R³. Then, by Stoke's theorem, we would have
\int_{S^2}B\cdot\hat{r}dA=\int_{S^2}\mbox{curl}(A)\cdot\hat{r}dA=\int_{\partial S^2}A\cdot dl = 0
(because the sphere has no boundary: \partial S^2=\emptyset). But, on the other hand, a direct calculation using spherical coordinates gives
\int_{S^2}B\cdot\hat{r}dA=\int_0^{2\pi}\int_0^{\pi}\sin(\phi)d\phi d\theta = 4\pi
This is a contradiction that shows that B is not the curl of any vector field A.