The privileges listed in the documentation are not enough to successfully start a task in the ECS cluster.

The resources "arn:aws:ecs:<region>:<account-id>:task-definition/jenkins-slave:*" must be added. For example:

{
"Version": "2012-10-17",
"Statement": [

{ "Sid": "Stmt1452746887373", "Action": [ "ecs:DescribeClusters", "ecs:RunTask", "ecs:StopTask" ], "Effect": "Allow", "Resource": [ "arn:aws:ecs:<region>:<accountId>:cluster/<clusterName>", "arn:aws:ecs:<region>:<accountId>:task-definition/jenkins-slave:*" ] }

,

{ "Sid": "Stmt1452779658000", "Effect": "Allow", "Action": [ "ecs:RegisterTaskDefinition", "ecs:ListClusters" ], "Resource": [ "*" ] }

]
}