Let $X$ be a separable Banach space endowed with a non-degenerate centered Gaussian measure $μ$ and let $w$ be a positive function on $X$ such that $w\in W^{1,s}(X,μ)$ and $\log w\in W^{1,t}(X,μ)$ for some $s>1$ and $t>s'$. In the present paper we introduce and study Sobolev spaces with respect to the weighted Gaussian measure $ν:=wμ$. We obtain results regarding the divergence operator (i.e. the adjoint in $L^2$ of the gradient operator along the Cameron--Martin space) and the trace of Sobolev functions on hypersurfaces $\{x\in X\,|\, G(x) = 0\}$, where $G$ is a suitable version of a Sobolev function.