Let \(\Omega \) be an open, simply connected, and bounded region in \(\mathbb R^{d }, d \geq 2\), and assume its boundary \(\partial\Omega\) is smooth. Consider solving an elliptic partial differential equation \(Lu = f\) over \(\Omega \) with zero Dirichlet boundary values. The problem is converted to an equivalent elliptic problem over the unit ball \(B\); and then a spectral Galerkin method is used to create a convergent sequence of multivariate polynomials \(u _{n }\) of degree \(\leq n\) that is convergent to \(u\). The transformation from \(\Omega \) to \(B\) requires a special analytical calculation for its implementation. With sufficiently smooth problem parameters, the method is shown to be rapidly convergent. For \(u\in C^{\infty}( \overline{\Omega})\) and assuming \(\partial\Omega\) is a \(C ^{ \infty }\) boundary, the convergence of \(\|u-u_{n}\|_{H^{1}}\) to zero is faster than any power of \(1/n\). Numerical examples in \(\mathbb R^{2}\) and \(\mathbb R^{3}\) experimentally show an exponential rate of convergence.