Use a script task.
I just tried this. I created a File connection manager, with the connection string pointing to the file I wanted to write to. I then created a string variable containing the text to write.
I added a Script Task, specified my string variable in the Read Only Variables list, then clicked Edit Script. The script was as follows:
public void Main()
{
ConnectionManager cm = Dts.Connections["File.tmp"];
var path = cm.ConnectionString;
var textToWrite = (string)Dts.Variables["User::StringVariable"].Value;
System.IO.File.WriteAllText(path, textToWrite);
Dts.TaskResult = (int)ScriptResults.Success;
}
This worked with no problems.