Norwegian (aka Norwegian Air Shuttle) is launching new nonstop 787 service from JFK to Oslo and JFK to Stockholm in May, and they have some crazy low introductory fares. You’re looking at about $350 round trip (give or take, based on the day), taxes included (assuming you can get their incredibly temperamental website to load). Unfortunately you can’t earn miles on other programs, but really – for $350 you’re complaining?
Low fares are available to connecting cities as well (their website is torturing me – basically you fly to Oslo or Stockholm and you can connect across their lowfare routes in Europe). Children 2-11 get 15% off.
You can search for fares by month (fares seem to only display in Euro) to find the cheapest days.