I use vrbo.com, homeaway.com and airbnb.com for apartments. Many here also recommend vacationinparis.com.
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Since you are asking for neighborhood recommendations, I will assume this is your first visit to Paris. I like the 4th (Marais), 5th (Latin Quarter) and 6th (St. Germain) Arrondissments for their central locations. The closer to the river the better.
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Maybe people can recommend specific apartments if you can let us know how long you need it, how many people and what your nightly budget is in euros. Many apartments have a 3-5 night minimum.