We used talparis@hotmail.com to rent an apartment in Paris in April 2012. The address for the apartment is 16 Rue JJ Rousseau 75001 Paris. You can look it up on mapquest. I do not know the location as to 6th, etc. It was very close to the Louvre and Metro stops.

My wife and I enjoyed the experience. Our apartment was clean and modern. The pictures that they sent on the internet matched the apartment when we arrived. I think it was around $200 per night. You may want to email the owner for more information.

I used Parisbestlodge and loved dealing with them. They meet you at the apartment, they have a reasonable deposit policy, they are well reviewed and recommended by anyone who has ever used them( which is how I found them, through many good reviews on other travel forums ( and some here too) .

Their rates are reasonable and their apartments are exactly how they appear in photos( i was able to visit 3 others in my building and they all were exactly as portrayed).

A 75001 address is in the 1st arrondissement.

This thread has a lot of ideas for Paris apartments:

http://www.fodors.com/community/europe/paris-apartment-thread.cfm

We've used Paris Perfect for both ourselves and our clients. They are hard to beat since they own most of the apartments they represent(www.parisperfect.com/)

In September we rented the following 1 Bedroom VRBO apartment located in the 6th (see my 10/09/12 review under initials HHN). Sounds like it might be a good choice for you based on your requirements - ideal location (Rue des Grands Augustins), price, cleanliness, quiet, excellent local manager with prompt response to emails, comfortable beds - queen in bedroom and living room sofa that converts to a twin. We considered many apartments (primarily in the 6th) before choosing this one and were very pleased with the rental.
http://www.vrbo.com/359685

I use Vacation in Paris you pay in advance and get the keys befroe you leave home They taks cresit cards and have on site help if needed

Another vote for Parisbestlodge. Tierry is reliable and pleasant and the apartments are exactly as advertised.
We stayed a week in one in the 5th in September, that was just fine.