I am looking to rent an apartment in central Paris for one week in
September. My requirements are that the apartment is modern, bright
and well located for sightseeing, shopping and transport links.

It should be one bedroom, modern/trendy/stylish, well maintained and
have adequete cooking facilities.

My budget for the week is £800, $1500, 1100Euros.

Hi, take a look at the paris vacation rentals at
http://www.tiptopvillas.com/popular_searches/towns/206.php

I think this should have what you are looking for.