Domain Mapping for single site

Hi There,

Maybe it is a stupid question but we have the following situation with a client:
Our client has registered a domain with one service provider (a) and the SINGLE WordPress website is hosted with another hoster (b) and has its own IP-Address.

When we now change the IP-Adresse with (a) will the server (b) (and WordPress) automatically be reachable with und the new domain or do we have to use the domain mapping plugin to ensure that everything runs well?

Thanks for clarifying.