After lots of searching, debugging and head scratching I have gotten my DFRobot Ethernet Shield to work with my older Arduino NG board. The problem lies with a strategically placed resistor/LED combination on Pin 13 of the Arduino board. These components affect the SPI communication on Pin 13. To fix the problem you simply need to bypass the LED. See this forum post (Reply #7 by kg4wsv) and specifically this image. Thanks to LadyADA for pointing me in the right direction.
The problem area lies between R7 and R 13. Simply rotate R13 so that it connects to the other side of R7 (bypassing the LED).