Development/Migration to WordPress Site

Hi all, I’d like to re-design an existing site using WordPress but I don’t want to take the old site down while the new one is under construction.

What’s the best way to do this? I was going to build it on my own web server using one of my own domain names that is currently not being used. Once it’s complete, I was going to move it over to my client’s web server to replace his existing site.

Does this sound like a reasonable plan? If so, are there any pitfalls I need to be aware of. I’ve read some articles and it looks like moving a site AND changing the URL can get quite complicated. Is there perhaps a better way of doing it?

Any help would be greatly appreciated.