$1000 in midsummer for a transatlantic nonstop on a major carrier is a pretty good deal. From the west coast last summer, London fares bnriefly surged over $2000 in Economy.
If you're unconvinced you could try Icelandair via Reykjavik. Or fly into Amsterdam, usually the cheapest Eurohub from the States, and take the train down to Paris... it's three or four hours to the Gare du Nord.