If it's a simple static site, it shouldn't be much of a problem, just have the old server still serving the site for 48-72 hours after you change the dns, like Mr Yum suggested. That way users who are still directed to the old server are able to see the site.
If it's a dynamic site with forums and such, then it's a little bit more complicated but perfectly doable. My admin transferred my largest site today with huge forums (over 15k registered users and lots of active posting). There was zero downtime in the process.
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