Yup, downtime during a change of hosts is unacceptable, it is not such a difficult process for a host.
In a nutshell, upload your site to the new server, check it out, then, if you have access to DNS zone modification, change the IP of the domain(s) to point to the new server instead of the old. Your old nameservers will then send traffic to the new server. This way traffic will go to the new server immediately, unless you have the old domain cached (which usually never lasts longer than an hour or so). Then switch nameservers, and dump the old server after about 2 days or so.
There will be no downtime if you do it this way.
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Paravirtualization & Hosting since 2008
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