When we went to Paris in 2001 we were going to be there for 9 days so we rented an apartment.  That turned out to be such a good idea that we did it again this year even though we were only in Paris for 4 days.  The apartment we found through the internet was in this building (208 Rue de Rivoli) which has a traditional French cage elevator.  Tres chic. 

Directly across the street is the entrance to the Metro (the image is the Assemblée National station, not Tuileries) and the Jardin de Tuileries.  Tres convenient.

See the brass plaque high on the wall to the right of the door?  Click on it.