I've never rented an apartment in Paris but my instinct is that it might be challenging to find one for only a week (and mid-week to mid-week at that).
Do you need to be in downtown Paris? A hotel out in the 'burbs won't be terribly expensive. It might not be anything special either mind you but it could be adequate.
Example:
http://www.accorhotels.com/gb/hotel-...on/index.shtml
When I travel personally to Paris I like staying at the
Tim Hotel (Opera Madeleine). It's nothing special but I like the location and they have a rather decent breakfast there too.