Robert_Charlton - 9:05 pm on Feb 24, 2013 (gmt 0)
When switching hosts and not changing domain names, I always keep the original site up until DNS has fully propagated. Google and users should see one site or the other, but not both.
If your .htaccess is set up properly, Google won't index the site by IP number, so that should not be a problem. I've kept an original site up for a month on the old host before taking it down with no apparent difficulties. Probably not much point these days keeping it up longer than a week.
The above is a simplified description for a static site, and assumes you've got access to the A-records of your domain. I never host my DNS with my web host, as that can create propagation problems. Some web hosts make it difficult to leave them. On a site with dynamic content, you also need to pay attention to keeping the data on both sites in sync.