Question: I have a Fargate Service running in AWS. I use it to run multiple tasks. Some of the tasks connect to an RDS database to query the database. 

How can I add the Fargate Service to my inboard rules of a Security Group for the RDS database? - Is there a way to associate an Elastic IP with the Fargate Cluster?

Answer: 
Might have misunderstood something here... But the ECS allows you specify a security group at the service level. 

Go to <https://docs.aws.amazon.com/cli/latest/reference/ecs/create-service.html>
And search for the `--network-configuration` parameter 

So surely you just need to set the source on your inbound rule of the RDS security group to be that security group ID?
Fargate doesn't support associating Elastic IPs with clusters. Clusters which runs in Fargate mode operate on instances which are not yours, it's the opposite of classic ECS stacks. That means you can't manage networking of host instances.

There is a way to associate IP with stack by having a Network Load Balancer in front of cluster. Then you could add a rule which allows connect your cluster through NLB.