Icelandair has round-trip flights from New York (JFK) to Reykjavik (KEF) for $331-$358, NONSTOP. Flights January, February, April, and May. Departures on Monday, Tuesday, Wednesday, Saturday, and Sunday.
Some small OTAs on Skyscanner and Kayak have this for 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.
Icelandair offers free stopovers, which good if you want spend a few nights in Reykjavik. More details are on their website (link).
Click for more details and booking links Nonstop Flights: New York to Reykjavik $331-$358 r/t [Jan-Feb, Apr-May] – Icelandair / Kayak / Skyscanner