Shana tova umetuka! (Happy Jewish New Year!)
Ruby, 192 181 165 characters
require'date'
s=(-6+235*n=gets.to_i)/19*765433+73848;e=s/u=25920
$&gt;&lt;&lt;Date.jd(347641+e+([h=s%u/19440,84&gt;&gt;e%7+h&amp;1,v=e%7*(374454&gt;&gt;n%19&amp;1-h&amp;s%u/9923),v,n-5765].count 1))

Input is given as a single number on STDIN.
Edit 1: Several arithmetic transformations saved quite a few characters.
Edit 2: There was some room for improvement.