I've encountered this before. Same place I think for the same mission. However sadly I don't recall what I did to fix it, or if it somehow resolved itself the when I tried it again. Are you getting this EVERY time? Your game just locking up and freezing your PC?
Just a thought but maybe its something in your inventory that when "removed" locks up the game? Maybe give melburn your stuff and then store the rest somewhere else so Delphine takes nothing from you.
And is this the only time you take a carriage? Or do you take one often and this is the only time it locks up? It sounds like something isn't loading and you can't map from Solitude to the embassy through a loading screen.
Do you run the unofficial patches?