The Antarctic Impulsive Transient Antenna (ANITA) completed its second long-duration balloon flight in January 2009, with 31 days aloft (28.5 live days) over Antarctica. ANITA searches for impulsive coherent radio Cherenkov emission from 200 to 1200 MHz, arising from the Askaryan charge excess in ultra-high energy neutrino-induced cascades within Antarctic ice. This flight included significant improvements over the first flight in the payload sensitivity, efficiency, and a flight trajectory over deeper ice. Analysis of in-flight calibration pulses from surface and sub-surface locations verifies the expected sensitivity. In a blind analysis, we find 2 surviving events on a background, mostly anthropogenic, of 0.97+-0.42 events. We set the strongest limit to date for 1-1000 EeV cosmic neutrinos, excluding several current cosmogenic neutrino models.