Arduino Ethernet Shield (W5100) Cannot Access Outside Network

pluggy:
You don't have gateway / router IP address in your sketch. Without it, getting out can be very hit and miss because the arduino doesn't know where the internet is.

byte gateway[] = { 192, 168, 1, 1 };   //your router's IP address

After resetting it about 10 times and changing the IP address, zoomcat's code now works, though I'm not sure how consistent it will be.