IMHO Mike knew the Vatican Bank was in deep trouble. He first sets up the Vito Andolini Corleone Foundation and puts 100 million into the Vatican Bank's hands. This gets his foot in the door. With the first 100 million, there was already a tacit understanding between Gilday and Harrison that the Corleones were going to make a move to take over Immobiliare. At the party in Mike's apt. Harrison makes some kind of comment to Gilday about the family having increased its position and that he was worried. Gilday reassures him by saying they already have "an arrangement." Then Michael goes to Gilday and they negotiate how much it will take to get the Vatican Bank to cast its vote to give Corleone control over Immobiliare. He ends up depositing 600 million in the Vatican bank, and thus keeps it from going under. This is nothing more than a bribe to get the vote he needs. The swindle happens in Rome. The deal was always subject to ratification by the Pope, but the Pope died, and therefore Michael was left with a minority share in Immobiliare and 700 million sitting in a bankrupt bank. On the advice of Don Tomassino he then goes to Cardinal Lamberto to tell him the whole sordid story, and of course Lamberto becomes Pope and he does ratify the deal (we learn this during the opera).

One question that has always been troublesome, however is whether or not Immobiliare was really "legitimate." At the meeting in Atlantic City all the Dons are grumbling about how they want to wet their beaks in Immobiliare as well, something Michael refuses to do. During that scene one of the Dons makes a comment about how Immobiliare is laundering money in various countries. If that was common knowledge, then it is dubious that Michael really thought Immobiliare was entirely on the up and up.