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Point DNS of the second domain name to host.
Send out link request change to all sites pointing to old domain letting them know the domain will change.
All pages at your site add the new domain name to a link anchor.
Google will superimpose links as it spiders and once getting wise to the change will automatically shift/split to the new domain.
You can monitor your backlinks to see which pages (internal/external) are not being credited to the new domain.
If all links on your site are relative links (not including old domain name) you will have zero ill affect.
Once all external links have changed to indicate new domain... you can dispose of the old if you wish.
Imagine the shared server scenario with many different sites all using the same IP address.
The key there is "many" not "two".
Government sites (tourism in particular) do this all the time. They are stuck with a gov URL but add a brand domain.
The backlinks of the brand eventually takeover the site and the gov eventually only has a root listing.
<added>check microsoft.com & microsoft.net backlinks they are both on the same IP's</added>
the problem was tracked to deep linking where somebody had linked to www. whatever .co.uk/somepage.htm the dns still pointed the co.uk to the main folder and the page still exists so google saw a .co.uk and a .com showing dupe content the page was dropped for three months.
in the end we had to setup 302 and 301 on all the pages which had been deep linked and renamed the new to n_oldpagename.
My original answer was maybe to short but webdev asked what would happen if I changed from a to b not how to change with the least damage
DaveN
The more likely scenerio would be the deeplink was a higher level page with many internal links off it.
As the site is credited to the new domain "but a single deeplink" draws page ownership to old domain - this separates lower level pages from a coherent link hierarchy. This smaller (one page domain) stuck between Top Level and Lower Level pages make the new domain appear much smaller.
External link requests first resolves this problem.