How do I re-design a site without affecting the live published version?

Hi all,

I am helping to re-design a live current site. The owners are wanting a completely different look, so this entails me essentially starting from scratch. They want to change their domain name, also. How do I undergo intense edits like these without messing up the current state of the live site? I don’t want to publish all of my edits until the new site is essentially complete.

The page I need help with: [log in to see the link]