Apartment is Sarlat

For an apartment in many places in Europe, we've had great success with HomeAway (www.HomeAway.com) and their sister company VRBO (www.vrbo.com).

For an apartment (and/or a gite) specifically in France, our first place to look is always Gites de France (http://en.gites-de-france.com). One quirk is that their places always rent out strictly Saturday to Saturday, so if you're arriving in Sarlat on a day other than Saturday, you'd need to make other lodging arrangements until that next Saturday arrives. On the Website, make sure that the English flag appears in the upper right-hand corner (unless you're fluent in French), then click on the Aquitaine section in the southwestern portion of the map of France, then click on the Dordogne portion of the map that comes up. You can search for accommodations by size and amenities.

We've almost always used Gites de France, but for an upcoming trip to the Dordogne we used one of the tourist bureaus. I think it was one of these:
http://www.dordogne-perigord-tourisme.fr/
http://www.perigord-reservation.com/fr/index.htm
http://www.sarlat-tourisme.com/