I'm trying to fit a model of the form $Y=aX+b$ based on a number of $(X,Y)$ observations with non-independent errors in $Y$. I know the variance-covariance matrix of the errors on $Y$.

1. How can I compute best-fit parameters $(a,b)$ and their var-cov matrix?
2. Can a least squares regression take non-independent errors into account?
3. Is there a more classical method to solve this kind of problem?