Are the domain name and hosting supplied by different companies?
Typically, a domain name will be held and paid for at a domain registrar. Within the domain control panel will be settings for nameservers and DNS.
The domain will likely be using the registrars nameservers. Even if you discontinue using any hosting supplied by the same company, you can still leave the domain there using their nameserver, and just update the DNS records to point the website at the new hosting. A particular part of the DNS records for the domain called MX records decide which mail servers (in your case google mail servers) email gets directed towards. As long as you don't touch the MX records or change nameservers, the email should be safe.