I found a simple formula in Numerical Recipies in C [page 157, Eq. (4.5.27)] to compute the weights for quadratures where the orthogonal polynomials can be given with a three-term recurrence expression. This formula is the following:
$w_j = \mu_0 v_{j,1}^2$, (*)
where $v_{j,1}$ is the first component in the $j$th eigenvector of the (symmetrized tridiagonal) Jacobi matrix, and
$\mu_0 = \int\limits_a^b W(x) dx$
with a given weight function.
Behind this formula, it is supposed (?) that $a$ and $b$ are the integration limits of the individual quadratures (\eg, $a = -1$ and $b = 1$ for the Gauss-Hermite quadrature). How should the (*) equation be modified if the integration limits $a'$ and $b'$ are different from $a$ and $b$, respectively?