If Google continue to list an ‘old’ version of your site (i.e. to list www.my123site.org despite the fact that your site now lives at www.my456site.org ) you need to update the links that are pointing to the sites.
Since Google robots jump from page to page via hyperlinks, someone must still be linking to the defunct page. Once others correct their links, Google can too.
Once your new site is live, you may wish to place a permanent redirect (using a “301” code in HTTP headers) on your old site to inform visitors and search engines that your site has moved.
Excerpt taken from Google Webmaster Info