Help with HTML Site Makeover to WordPress

I’m redesigning a site for someone that currently has a static HTML site that is hosted by Web.com. This company that I’m building the WordPress site for wants to keep the same URL (hosted by another company) that they currently have and they don’t want any down time on their website as I build the new site.

I’ve only built WP sites from scratch where I put the domain name in my hosting system and install WordPress to it.

Can this be done another way?