Icelandair has round-trip flights from New York (JFK) to Reykjavik (KEF) for $347-$358, NONSTOP. Flights January to May. Departures on Monday, Tuesday, Wednesday, Saturday, and Sunday.
Some small OTAs on Skyscanner and Google have this for as low as $315; which is cheaper than the airline. If you plan on booking there, beware of any site that charges a fee to cancel a ticket within the 24-hour “free” period. Use unknown booking sites at your own risk.
A Carry on bag (20-lbs @ 21.6 x 15.7 x 7.8 inches) and a personal item (15.7 x 11.8 x 5.9 inches) is allowed for free on Icelandair. You can avoid seat selection fees by accepting the randomly assigned seat at the gate.