Thanks to some help from people on this forum I have fixed the last couple of issues and am now able to release the first version of the SerialIP library.
It's an Arduino-specific wrapper around the uIP library, which implements a TCP/IP stack in software. It works by establishing a SLIP connection over the serial port to your PC. (Remember SLIP? It was used in the early days of dial up Internet.)
Once you have the SLIP connection established you can communicate with the Arduino like any other network device. It will respond to pings and the library currently includes two examples - one that runs a simple server and one that sends an e-mail.
There are instructions and a download link on the wiki.
I've developed it under Linux, but unfortunately I don't have enough experience with Windows or Mac to know how to establish a SLIP connection under these operating systems. If anyone knows this please update the wiki page so that others can make use of the library.
I hope someone finds this useful!