Rebuilding a site on a new host and with WP. Best method?

So I have a client with a domain and site hosted on godaddy. They hate it and want me to make them a WP site and host it on my reseller account.

What is the best method to do this?

I was thinking I create the account with the current domain name then install WP and build site, when its done, then change the DNS nameservers to point at the new hosting. There must be a way to install WP using just the IP?

Or should I build it on a spare domain I have, get it done and then move the whole site to a new account after I have pointed the domain? Though this seems like it would have down time.

Usually I am designing fresh new sites and this isn't an issue and when I agreed to the gig I forgot to consider this scenario.

First transfer the domain to namecheap or something other than godaddy (because godaddy supports SOPA and can be difficult to work with at times). Then build the site on your spare domain and do a 301 redirect to your your spare domain while you wait for godaddy to transfer the domain. Once the transfer is complete change the DNS nameservers to your reseller hosting account.

Yeah, not my call on the transfer of a domain to namecheap, just paid to build a new site and move it to my hosting.

Build the site on a demo domain not the main domain, when everything is good use BackupBuddy to move the whole site to his new account I do this to all my clients, backupbuddy is really useful.