  In a round ball $B\subset\mathbb{R}^{n+1}$ endowed with an $O(n+1)$-invariant
metric we consider a radial function that weights volume and area. We prove
that a compact two-sided hypersurface in $B$ which is stable capillary in
weighted sense and symmetric about some line containing the center of $B$ is
homeomorphic to a closed $n$-dimensional disk. When combined with Hsiang
symmetrization and other stability results this allows to deduce that the
interior boundary of any isoperimetric region in $B$ for the Gaussian weight is
a closed $n$-disk of revolution. For $n=2$ we also show that a compact weighted
stable capillary surface in $B$ of genus 0 is a closed disk of revolution.
