The most common proof that I have found of the fact that Ampère's law is entailed by the Biot-Savart law uses the fact that, if $\boldsymbol{J}:\mathbb{R}^3\to\mathbb{R}^3$, $\boldsymbol{J}\in C_c^2(\mathbb{R}^3)$, is a compactly supported twice continuously differentiable field such that $\nabla\cdot\boldsymbol{J}\equiv 0 $ and $\Sigma$ is a smooth surface satisfying the assumptions of Stokes' theorem, then $$\oint_{\partial^+ \Sigma}\left(\frac{\mu_0}{4\pi}\int_{\mathbb{R}^3}\frac{\boldsymbol{J}(\boldsymbol{x})\times(\boldsymbol{r}-\boldsymbol{x})}{\|\boldsymbol{r}-\boldsymbol{x}\|^3}d\mu_{\boldsymbol{x}}\right)\cdot d\boldsymbol{r}=\mu_0\int_\Sigma \boldsymbol{J}\cdot\boldsymbol{N}_e \,d\sigma\quad(1)$$where $\mu_0$ is any constant (the magnetic permeability in the physical interpretation), $\boldsymbol{N}_e$ is the surface's external normal unit vector and $\mu_{\boldsymbol{x}}$ is Lebesgue $3$-dimensional measure.
Nevertheless, common exercises and applications of Ampère's law found in books of physics use current densities $\boldsymbol{J}\notin C_c^2(\mathbb{R}^3)$, one example being $\boldsymbol{J}$ constant on an infinite cylinder and constantly $\mathbf{0}$ outside the infinite cylinder. Do mathematically rigourous formulations of Ampère's law $(1)$ exist under more relaxed assumptions on $\boldsymbol{J}$, like the quoted case of $\boldsymbol{J}$ constant on a (bounded or unbounded) region and null outside of it, and, if they do, how can they be proved? I have thought about approximating such a $\boldsymbol{J}$ with $\boldsymbol{J}_n\in C_c^2(\mathbb{R}^3)$, but it is not easy to see that the required sequence really exists. I heartily thank any answerer!