I've had to help many people solve this problem, so I thought it might be useful information for macosxhints.com. The way to solve the problem is to delete related cookies:
- In Safari, go to Preferences » Security » Show Cookies.
- Type in live.com
- Click the Remove All button.
[robg adds: A comment on the queue site notes that this should also help fix looping issues with Yahoo's websites. I'm not a user of either HotMail or Yahoo (and Safari isn't my main browser), so I can't say I've experienced any of these issues. If you have, and this hint helps, please note it in the comments.]

