AWS API Gateway No 'Access-Control-Allow-Origin' header is present

Digitalkapitaen's answer is correct; here is the code to save someone the trouble of looking up how to set an HTTP response header in Lambda:

exports.handler = function(event, context, callback) {
    callback(null, {
        "statusCode": 200,
        "headers": { 
            "Access-Control-Allow-Origin": "*" 
        }
    });
};

The response for the GET request to the Lambda function must also contain the Access-Control-Allow-Originheader.