The Doom III demo should be "coming in a few days" according to id Software programmer Robert Duffy. It's just "finishing the last stages of QA" at the moment. In a recent .plan update, Duffy also spoke of the contents of the demo: "It contains three levels of the game and is single-player only," he wrote.
In the meantime, id Software has released a beta version of a version 1.05 patch for Doom III, fixing a host of niggling problems. "We don't anticipate any issues with it but felt it best to get it into the hands of a few thousand people rather than the entire customer base in case we missed something," said Duffy, directing willing guinea pigs here for the file.
"We will release the update through our update server for wide release once we have verified there are no problems with it," he added, also promising another update would be made available soon with Punkbuster anti-cheat functionality, and that the Software Development Kit (SDK) and Linux dedicated server are both ready to go and will be released "as soon as the update is out in non beta form". You can find Duffy's complete update (including a list of changes the patch makes) here.