The pope's bank has held its first-ever press visit – but television cameras were forbidden and journalists had to submit questions in advance

On the ceiling of the conference room in the Vatican bank, there is a big, round allegorical painting.

There is the Virgin Mary somewhat improbably wearing a papal tiara and holding a model of a church. There is another holy lady offering the mother of Jesus a gold plate laden with crowns, a gold chain and an honorific decoration. In the background, Neptune is emerging from the sea on a chariot, while in the foreground there is snake winding its way through a patch of mushrooms.

Baffled? Well, what did you expect? This is the heart of what, until Thursday, was the most secret building in the "city of secrets" – the premises of an institution that has attracted more speculation, suspicion and sinister insinuation than almost any on earth.

Addressing a restricted number of correspondents on the first-ever press visit to the Vatican bank, its managing director, Paolo Cipriani, could scarcely be expected to recall its involvement with Banco Ambrosiano – whose boss, Roberto Calvi, was found lifeless and dangling by the neck under Blackfriars Bridge in 1982 – or the claim by Holocaust survivors that the bank took in gold looted by wartime Croatian fascists. He merely alluded briskly to "that veil, that shadow that comes from the past", and then plunged back into an account of the bank's efforts to satisfy international requirements on transparency.

Next month, a committee of the Council of Europe is to report on the Vatican's progress towards compliance with rules designed to prevent money laundering and the funding of terrorism. The aim of the briefing and tour was to persuade the world that the bank – or Institute for the Works of Religion (IOR), to give it its proper name – was as clean as a new white altar cloth.

Not the least of the IOR's handicaps is that it is housed in what was once the papal prison – a grim, grey tower built in the 15th century and attached to one side of the Palace of Sixtus V, the pope's residence. We entered through a walled courtyard at the side and were taken up five flights of stairs into the conference room, through a passageway in which old ledgers were on display together with two magnificent coin collections.

After the briefing, we were directed along a corridor to a winding staircase that led down though the tower and eventually delivered us into a big, circular hall. Ranged across one side were the tellers' positions. Light entered the room through high, broad windows let into the massively thick walls of the former jail.

At the bottom of a flight of stairs at the entrance was an ATM. In Latin, the opening screen announced it was available for a cash withdrawal – a "deductio ex pecunia".

According to Cipriani, the IOR has 33,000 accounts belonging to clerics, religious orders and foundations, Vatican employees and diplomats accredited to the Holy See. He said the IOR had used scrupulous checks to prevent them being used by third parties.

Openness at the Vatican usually comes in limited doses, and this was no exception. TV cameras and recording devices were banned and questions for Cipriani after his presentation had to be submitted in advance.

But the IOR could scarcely find a more reassuring face to turn to the outside world than the bespectacled and ever-so-slightly chubby one of its genial managing director. He admitted that he had found it difficult to adapt to the ways of the Vatican after a lifetime spent in commercial banking. He ended by draping across the front of his lectern a T-shirt he had been given by another banker with "Anti-money Laundering Expert" and a representation of a washtub emblazoned on the front.

At one point during his presentation, he clapped his hand to his mouth after saying nuns could not open accounts "because they have taken vows of chastity". Looking just like a choirboy heard using rude words in the vestry, he said: "I meant 'poverty'."

Not all of the shadows cast over the IOR, however, date from its distant past. In 2010, Cipriani and his then president were put under investigation in Italy for a suspected violation of anti-money-laundering laws that the Vatican said was due to a misunderstanding. In February, JP Morgan broke off a 35-year relationship with the bank after the IOR refused to furnish it with information the Vatican considered excessive. And three months later, the president, Ettore Gotti Tedeschi, who had been brought in specifically to improve transparency, was suddenly dismissed.

Cipriani gave the Vatican's side of both the first two controversies. But, in a presentation lasting some two and a half hours, Gotti Tedeschi's name was not mentioned even once.