You must have a registered internet domain name in order to set up custom domain names for your APIs. If needed, you can register an internet domain using Amazon Route 53 or using a third-party domain registrar of your choice. An API's custom domain name can be the name of a subdomain or the root domain (also known as "zone apex") of a registered internet domain.
After a custom domain name is created in API Gateway, you must create or update your DNS provider's resource record to map to your API endpoint. Without such a mapping, API requests bound for the custom domain name cannot reach API Gateway.
Take a look at this link which walks you through setting up custom domain names for REST APIs
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