Custom domain for API Gateway using CloudFormation

Question

I'm trying to define a custom domain (sub domain actually) for an API Gateway using CloudFormation. According to all the documentation I've been able to find the following should work but when I deploy it I get Invalid domain name identifier specified for the BasePathMapping

aws cloudformation deploy --template-file packaged-prompt.yaml --capabilities CAPABILITY_IAM --stack-name prompt-stack

prompt.yaml

AWSTemplateFormatVersion: '2010-09-09'
Transform: AWS::Serverless-2016-10-31

Resources:

  ...

  Certificate:
    Type: 'AWS::CertificateManager::Certificate'
    Properties:
      DomainName: example.mydomain.com

  DomainName:
    Type: 'AWS::ApiGateway::DomainName'
    Properties:
      CertificateArn: !Ref Certificate
      DomainName: example.mydomain.com

  Mapping:
    Type: 'AWS::ApiGateway::BasePathMapping'
    Properties:
      DomainName: !Ref DomainName
      RestApiId: !Ref Api

  Api:
    Type: AWS::Serverless::Api
    Properties:
      StageName: prod
      DefinitionBody:
        swagger: 2.0
        info:
          title:
            Ref: AWS::StackName
        paths:

          ....

Answer 1

The properties in your BasePathMapping is missing

BasePath: ""

Answer 2

It may not be the case, but I had the same problem as I was using uppercase letters in my domain-name, which must be avoided. See BaseUrl Mapping in AWS with inconsistencies