Upvote:5
Olivier Bernier's biography 'Louis XIV' says, based on Mme de Motteville's account (II, 286):
"Having seen the Queen in her bed, we went off home ... As soon as we had left, the gates of the Palais Royal were closed with the command to not open them again. The Queen got up again to think about her situation and confided her secret only to her First Woman of the Bedchamber who slept near her ..."
"The necessary orders were then given to the captains of the guard ... The marΓ©chal de Villeroy allowed the King to sleep until three in the morning; then he roused him, along with [Monsieur -- in reality, the duc d'Anjou but simplified in Bernier's account], and brought them to a carriage which was waiting for them at the garden gate of the Palais Royal. The Queen joined the King and the Monsieur."
From there, it was a very short ride to the Cours la Reine, safely outside the walls of Paris.
Previous movements of the court are not as precisely described. Bernier notes that the king had left Paris for Richelieu's palace on September 12th, and then returned to "Paris" two days after October 22nd.
The only option for confusion could be if the King was brought to the gate of the Palais Royal from the Louvre though nothing in the above text suggests that; I'd recommend Mme de Motteville's personal account for that (no doubt easier accessed in French -- I didn't find an accessible translation at present).