Update February 8, 2012: According to a JFK-BUD Suspension of Service notice on aa.com, AA non-stop flights between JFK and Budapest are no longer available for booking despite the previously planned service resumption. This cancellation likely comes in the wake of Malév's collapse, as American Airlines (currently under bankruptcy protection) is reorganizing some of its routes. AA continues to offer travel choices between the US & Budapest with oneworld partners, British Airways & Finnair, via either London (LHR) or Helsinki (HEL).
Update January 15, 2012: Beginning April 3rd (New York to Budapest) and April 4th (Budapest to New York) AA will resume it's non-stop flights. Tickets are on sale now.
Update November 21, 2011: AA decided to stop their daily non-stop flights during the winter season. In 2011 the last flight from New York to Budapest is scheduled for November 30 and the first flight in 2012 is scheduled for April. AA flights during the winter season operate via London and Paris.
More direct flights from New York to Budapest
According to the official press release (October 7, 2010) American Airlines will start direct flights between New York (JFK) and Budapest as of April 5th, 2011. The aircraft, a Boeing 767-300, will be operated by AA in a code-share agreement with Malev Hungarian Airlines. Both airlines are part of the oneworld alliance, which means that passengers will have access to each other's networks, fares and frequent flyer programs. Reservations can be made with both airlines.
Malev has had a long history of code-share and Malev operated direct flights between New York and Budapest. Scheduled service to New York with Boeing 767 aircraft began in May 1993 and operated until October 2008. Currently Delta has scheduled, non-stop flights between New York and Budapest until mid October. According to their schedule they will resume their service as of next summer. The fact that two US carriers will be competing on the same route should mean better rates and increased frequency for passengers.
Click here for some suggestions on how to find cheap flights to Budapest.