Change url without moving site

I need to change the URL that a Wordpress site uses and shown as in admin general settings.
I have a development site on a domain that is coming up to renewal. I do not want to use that domain now so have registered an alternative domain. I can easily map these domains together so that the server uses the new domain. I don't have to do a file transfer for this. Once it has updated on the server the domain that I don't need can be deleted and it will be as the site is assigned to the new domain.
Apart from... It will probably all fall over as the admin setting for the site address won't be right.
What is the right way to overcome this please?. I'm sure that just changing the domain address won't achieve the right desired result.
Sorry if this question comes across as simplistic but each time I have done this in the past it has failed and busted the site.
Regards