I have several old iWeb sites that I'm rebuilding in Sandvox because Apple is no longer updating iWeb and will apparently let it die a slow death. I once hosted on Apple's MobileMe, but those servers are going dark at the end of June 2012 as Apple completes its move to iCloud, which doesn't allow web hosting. Here's how I'm making the transition with these sites:
I publish my entire old iWeb site on my new host, A2 Hosting, along with my new Sandvox site. They both live comfortably on the same domain because all the URLs are different. For starters, and using the Original Green site as illustration, the Sandvox site is at the root of www.originalgreen.org, whereas iWeb required something they called a Site name. It amounted to a subdirectory, I guess. I chose OG as my Site name when I first built the Original Green site several years ago, meaning that all pages on the old site are at www.originalgreen.org/OG/. Also, iWeb and Sandvox use different page-naming conventions, so no two pages are named the same.
Building a site can be a big job, depending on how big and how complex it is. So when you finally publish the site, you probably have a lot of other things that have been piling up that you need to work on. There's one more piece of work left to complete the transition, but you can put it on the back burner if both sites are published as described here: it's the chore (there's no better way to put it) of creating redirects for each page on the old site. Once you've completely finished redirecting the pages, you can go to your host and delete the old site entirely. Just make sure that you double check that you've finished every page.