We consider the Stokes-transport system, a model for the evolution of an incompressible viscous fluid with inhomogeneous density. This equation was already known to be globally well-posed for any $L^1\cap L^\infty$ initial density with finite first moment in $\mathbf{R}^3$. We show that similar results hold on different domain types. We prove that the system is globally well-posed for $L^\infty$ initial data in bounded domains of $\mathbf{R}^2$ and $\mathbf{R}^3$ as well as in the infinite strip $\mathbf{R}\times(0,1)$. These results contrast with the ill-posedness of a similar problem, the incompressible porous medium equation, for which uniqueness is known to fail for such a density regularity.