You can access the archive of contemporary French newspapers using agencies such as Factiva or Nexis. Note: Le Monde can no longer be accessed through Factiva or Nexis.
The library of the French Institute has institutional access to Europresse, a French website that gives you access to 1500 online newspapers and magazines, among which 300 titles in French (and Le Monde of course). If you are a member of the French Institute, you can access this website for free, for individual use (but you will need to pay a subscription to join the Institut Français). Gallica has published a special number of its e-journal about its holdings of French newspapers.