The Chow Ring of the Hilbert Cube

Given a smooth projective variety $X$ over an algebraically closed field $k$, we compute the Chow ring of the Hilbert scheme of three points on $X$, ${\mathrm{Hilb}}^3(X)$, as an algebra with generators and relations over the Chow ring of $X\times {\mathrm{Sym}}^2(X)$. If in addition the characteristic of $k$ is zero, we extend the computation to the quasi-projective case.