This time something from tips & tricks with AWS Fargate. Setting up service we can add it as publicly available. Thus there is no option to assign EIP or domain as container can be stopped or restarted any time with new IP. The only one way to achieve it is usage of ELB (Application LB, EC2 service). You have to configure it to point your Fargate service to LB and then use  ELB public address (name, not IP), and register correlate with your domain in DNS as CNAME.