MacOS – Full-screen apps and desktop arrangements

What you can do is this:

Delete the desktops and then recreate the by hovering over them and pressing the red thing in the corner, then looking at the right to make a new desktop.

This is no longer necessary as of Mac OS Lion 10.7.2, see this answer for an easier solution.

Make sure you have "Allow Exposé, Dashboard and others to use the screen" in keynote preferences under Slideshow.

Then, before you slide over to Safari, press the "F" key to pause the slideshow. This prevents the slideshow from disappearing.

You can then slide over, or use Mission Control, etc without affecting the slideshow.

There is a bug here, to go back to the slideshow you have to slide back to it or click on the desktop in Mission Control; you can't click the Dock icon or Cmd-Tab back to it.
(Clicking the dock icon / cmd-tabbing will just hide the dock and menubar but not take you back to the presentation)

Also: When you get back to the slideshow, the first key press or mouse click unpauses the presentation, so you will have to click / press again to advance to the next slide.

Side note: instead of having keynote and Safari on separate desktops, you can have them both as fullscreen apps. This hides the Dock and Menubar, so when you show the web page, there is more room for it.