Verify domain ownership using Search Console
Go to the Google Search Console and Add property
In the Domain option (not URL Prefix) add the top-level domain, sometimes called the naked domain, without http:// or https:// prefixed to it (e.g. example.com) your site cannot use this address but you need to verify ownership of this entire domain.
Use the Instructions for: Any DNS Provider.
Copy the TXT record that Google Search Console gives you as a value to add to your DNS records.
Note:
You can assign up to 5 custom domains/ URLs
If you have published the site to only a specific group of people the custom URL will redirect to the sites.google.com URL
These are DNS records you need to add to make the custom URL point to your personal Google Site
Add a TXT record with a name/host/subdomain (whatever term your domain provider uses) of @ and the value/data/target use the text string copied from Google Search Console mentioned above
Add a CNAME record with a name/host/subdomain of www (or whatever subdomain you want to use) that points to ghs.googlehosted.com.
If your domain provider lets you redirect your naked domain to the subdomain address (e.g. make example.com redirect to www.example.com) then configure that
Add an A record of @ (or your naked domain) that points to 174.129.25.170 (a free service from wwwizer.com)
Once the top-level domain is verified (may take hours depending on how quickly your domain name provider updates its public DNS records) you can then add a subdomain of your domain as a custom URL to your new Google Site: follow the steps below to make this custom URL web address mapping.
Note:
You must verify your own custom DOMAIN (e.g. example.com) NOT your sites.google.com URL
You cannot use the top-level (naked) domain (e.g. example.com) as your domain provider will not allow you to make the root-level DNS record a CNAME record, so you must use a subdomain (e.g. www.example.com)
Use the Settings (looks like a cog/gear) near the Publish button
Choose the Custom URLs option
In the box labeled www.yourdomain.com enter your domain with www. prefixed to it, e.g. www.example.com
Close the Settings dialogue box.
Wait. it takes a little time to configure everything for your site at the back-end for the site will appear at the custom URL
Note:
You need to have published the site first before your custom URL can work