This is probably not a failure of the assumption of Gaussian arrors, though that may indeed be the case too. I cannot know.
What a singularity usually tells you is there is some linear combination of the parameters that does not change the sum of squares. You cannot obtain confidence intervals, since at least within some range, you can trade off any point for any other in the solution space along that line. In this case, the linear combination is a simple one.
The zero column of the Jacobian at the solution tells you that the line is one where one of the parameters is completely useless in the model. Changing it has absolutely no impact on the sum of squares. It essentially says you could fix that parameter at any value you wish. Perhaps zero is an option.
That you have an excellent fit says the model is probably adequate without needing that spurious parameter.
So remove the offending parameter from your model, as I said, fix it at your favorite value, if zero is not an option. Then redo the fit with one less parameter, and only then should you try to compute anything along the lines of a confidence interval.