Eurofly Airlines, a division of Meridiana airlines, will begin their seasonal direct-flight service from New York's JFK airport to Palermo (PMO) and Naples (NAP).  According to their site, the service will run from June 2011 through September 2011.

This will be the only direct flight from JFK to Palermo or Naples, unless another carrier follows suit. 

In September 2009, I took the JFK to Palermo route, and here is my review of Eurofly Airlines.