Robert_Charlton - 8:43 am on Sep 15, 2013 (gmt 0)
Tonearm - As you've laid it out, if you've vetted your new IP, a hosting change done properly shouldn't hurt your rankings at all.
JD_Toims is correct that it's absolutely important to have the site up and running on the new IP before moving, and also to keep an operating copy of the site on your old server during DNS propagation (and I suggest for a while after). Complexity of doing this depends on whether the site is dynamic or static and how frequently the data changes.
Assuming there are no problems keeping the site copies synced... if you are not changing domain name servers, then visitors and bots will see one version of the site or the other as DNS propagates, and the transition should be seamless.
If you are changing name servers along with your hosting, which is often the case, then TTL (Time To Live) settings for your DNS enter into it as well.
A "standard reference" that we've posted on and off here for many years is Matt Cutts' article on changing your web host while keeping the same domain name. It's generally been up in our "Hot Topics" area, which we haven't displayed for a while. Here's a direct link...
Moving to a new web host
I don't know the nature of your site and how important 24/7 operation is... but, IMO, it's much better to keep your hosting and your DNS servers separated, so if your hosting company has a problem, then you can easily get control of your DNS and switch to a backup copy of your site. This also allows the seamless propagation of DNS, as I note above.
You can either use your registrar for your A-records, which in some cases is adequate, or you might consider a dedicated DNS management service.
DNS fail-over to a backup site is also an option to consider. If you are needing to change the location of your domain name servers with this hosting change, this is a good time to consider your DNS management as well.