Changing custom primary vhost to another custom name

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I'm looking into a request to change a custom vhost for a main interface.  I read through Answer 2031 "Using SSL and your own domain name in the URL for a site" (https://cx.rightnow.com/app/answers/detail/a_id/2031) which talks about changing from custhelp.com to a custom vhost.  I understand this is done by first adding the custom vhost as an alternate then making it primary when the CNAME record is added and verified.  So how would the process work if we're changing the already custom primary vhost to another custom name?