In my mind I can even hear that call. And Bill saying to everyone else, "Why, Ah heard it on the news just like you did!"
I think that Hill/Huma/Bill schemed, Bill made the “here’s what we do” call to Weener, and Weener turns around and tells the papers that Bill or Huma informed him that Huma’s “pregnant,” (or Huma tells the papers—once she hears back from Bill that Weener has been briefed.)