要解决AWS ECS Fargate的出站互联网连接性问题，可以按照以下步骤进行：

1. 为Fargate任务创建VPC和子网。

2. 配置互联网网关（IGW）并将其附加到VPC。

3. 为任务配置NAT网关并将其放置在公共子网中，以便任务可以使用该网关进行出站互联网连接。

以下是一个示例CloudFormation模板，可用于自动化创建所需的VPC、子网、互联网网关和NAT网关：

Resources:
  MyVPC:
    Type: AWS::EC2::VPC
    Properties:
      CidrBlock: 10.0.0.0/16
  
  MyPublicSubnet:
    Type: AWS::EC2::Subnet
    Properties:
      VpcId: !Ref MyVPC
      CidrBlock: 10.0.0.0/24
      MapPublicIpOnLaunch: true
  
  MyPrivateSubnet:
    Type: AWS::EC2::Subnet
    Properties:
      VpcId: !Ref MyVPC
      CidrBlock: 10.0.1.0/24
  
  MyInternetGateway:
    Type: AWS::EC2::InternetGateway
  
  MyInternetGatewayAttachment:
    Type: AWS::EC2::VPCGatewayAttachment
    Properties:
      InternetGatewayId: !Ref MyInternetGateway
      VpcId: !Ref MyVPC
  
  MyNATGateway:
    Type: AWS::EC2::NatGateway
    Properties:
      SubnetId: !Ref MyPublicSubnet
      AllocationId: xxxx-xxxxx-x-x-x-x-x-x
  
  MyRouteTable:
    Type: AWS::EC2::RouteTable
    Properties:
      VpcId: !Ref MyVPC
  
  MyPublicRoute:
    Type: AWS::EC2::Route
    DependsOn: MyInternetGatewayAttachment
    Properties:
      DestinationCidrBlock: 0.0