See New Features Rolling Out Soon

Mapping to Domain

Your site must be published to a sites.google.com URL before it can be mapped to a domain.

From Google at https://support.google.com/sites/answer/9068867?hl=en-AU#zippy=%2Ccaveats

You can use a custom domain for a site published on new Google Sites.

Enter a site name in the 'Web address' box. Use only lowercase letters, numbers and dashes. Your site URL will appear below the box. Do NOT enter the domain!
If you want to use a custom domain that you own click Manage.
Alternately you can just publish the site and set up the custom domain later.

Set up a Custom Domain

Go to Settings.
Click Custom domains.
Click Start setup.
Select an option below and follow the additional steps.

Note: You can connect up to five custom domains to one site. It may take up to 48 hours for changes to your domain to become visible on the Internet.

Custom URL

Go to Settings.
Click Custom domains.
Click Start setup.
Select an option below and follow the additional steps.
Note: You can connect up to five custom domains to one site. It may take up to 48 hours for changes to your domain to become visible on the Internet.
Note: Only site owners will see Custom domains. Editors will not.

See Custom URL instructions for GoDaddy

For G Suites accounts read more here and see Steegle's instructions here .

For domains registered with Google Domains click here. Note that when a domain is registered at Google Domains, domain ownership proof is automatic. See also Using Workspace account.

If you want a domain link that goes to a particular page in your site, use forwarding instead of mapping. For instance rate.k8oms.net

I did this by setting up forwarding in my domain panel to forward the subdomain 'rate' to a page in the site. See the image below to see how I did it in Godaddy.

New Google sites are automatically secure (https) at the native Google URL and at the mapped domain, so it is not necessary to purchase a SSL certificate. If necessary, use this procedure to refresh the certificate: Remove the custom URL from your site, wait about 15 minutes then add the custom URL again and that will generate a new SSL certificate.

More information on Custom URLs

I don't see custom domains option

There are 2 reasons that you cannot see the custom domain option in Google Sites:

You use Google Workspace so you need to add the custom domain from the Google Workspace admin panel - see the link below:
- Map a Custom Domain URL (G Suite) - new Google Sites
If you do not use Google Workspace it means you're not signed in to the site as the site owner - only the site owner can add a custom domain, so sign out and back in as the owner.