We prove that the ring ℝ[M] of all polynomials defined on a real algebraic variety $M\subset {ℝ}^n$ is dense in the Hilbert space $L^2(M,e^{-{\left|x\right|}^2}d\mu )$, where dμ denotes the volume form of M and $d\nu =e^{-{\left|x\right|}^2}d\mu$ the Gaussian measure on M.