Can I do an integral, possibly using gaussian quadrature, when the abscissas are fixed (for reasons that I don't want to get into right now), i.e. is it possible to calculate the weights for fixed abscissas that I don't get to choose?
My interpretation of the problem: given $n$ pairs $(x_j, f(x_j)$ with $a \le x_1 < x_2 < \ldots < x_n \le b$, you want an approximation to $\int_a^b f(x)\, dx$.
One way, that would give the correct value for polynomials of degree $\le n-1$, would be to use $\int_a^b g(x)\, dx$ where $g$ is the Lagrange interpolating polynomial of degree $\le n-1$ corresponding to your data. However, that can be very unstable (see "Runge's phenonenon"). A better idea might be to use something like a composite quadrature rule: partition your interval into subintervals, each containing a few of the $x_j$, and use the interpolation for those $x_j$ in the subinterval to approximate the integral over that subinterval.
At the risk of being redundant, I'd like to mention here some other things I mentioned in the (now expanded) answer of mine at that other site for completeness' sake.
Robert's warning of the Runge phenomenon happening is a good one, and it does happen if your abscissas are perversely distributed (relatedly, the underlying Vandermonde matrix is ill-conditioned); the equispaced case being among the worst-behaved point distributions. Abscissas that are "nicely distributed" (e.g. Legendre, Chebyshev, or any other point distribution which "clusters" near the ends) will generally ensure that you have a quadrature rule that behaves tamely even for large numbers of points. (As an aside, the
chebfun project hinges on functions being nicely approximated by interpolating polynomials with abscissas that are transformed Chebyshev polynomial roots/extrema.)
Lastly, whatever you finally settle with, you will want to perform the sanity check of ensuring that all the weights of your quadrature rule are of the same sign; any change of sign in the weights can lead to subtractive cancellation when you employ the quadrature rule, and you wouldn't want that...