Five weeks ago, the fraudsters telephoned 65-year-old Nargess Sadjady, from west London, in a series of calls.
Posing as staff from her bank, they can be heard persuading her to transfer the money into one of their own accounts.
The recordings were set up by her son, who was worried that either of his parents might fall victim to just such a fraud.
He installed special software on their phone, and subsequently passed the recordings to the BBC.
Originally the police decided not to get involved, as they said there was no realistic prospect of a conviction.
But as a result of the recordings, to be broadcast on Radio 4's Moneybox programme on Saturday, detectives have now decided to launch a full investigation.
The first call to Mrs Sadjady was from a man calling himself Mark, who claimed to be from her bank, Santander.
He explains that he is worried about recent fraudulent transactions on her account.
He comforts her by saying they have managed to stop the payments, but they might not be able to stop subsequent attempts.
Mrs Sadjady can be heard trying to resist his enquiries.
"I don't know you," she says.
"Really I can't give more information. Please write to me, because I don't know you."
But the man calling himself Mark offers her layer upon layer of reassurance, including giving her a password.
"I'm not going to ask for personal information," he says.
Then he asks her to check the bank's helpline number on the back of her card, and tells her that she will receive a call back from that number.
In a trick known as "number spoofing", orchestrated through freely-available software, that number then appeared on Mrs Sadjady's phone when a second man called back, claiming to be from Santander's fraud department.
Later she admitted that up until this point in the call she had been sceptical - but the number spoofing persuaded her that it was genuine.
"I wasn't sure, until they called my mobile from that phone number on the back of my card," she said.
A few minutes later, the fraudsters made a third and final call. Sensing that she might be about to alert her bank, they persuade her that staff at her local branch may be responsible for the fraud.
So they suggest that she transfers £12,000 in her Santander account to a separate account at the Halifax, which they say has her name on it.
In reality they had already opened that account themselves. When money is transferred in, banks do not usually cross-check the names.
So, after one hour and 40 minutes of phone calls, spread out over three hours, Mrs Sadjady moved her money, voluntarily.
Almost immediately the fraudsters then started to distribute that money to other accounts in the UK, before the banks cottoned on.
It was a classic case of "Vishing" - or voice phishing - which last year cost consumers and banks £24m, according to Financial Fraud Action UK (FFAUK).
But it is thought to be the first time that such a crime has been recorded, and then broadcast.
"These scams are so sophisticated that people genuinely end up believing it is the bank, even though they know, potentially, that it doesn't sound quite right," said Tony Black, a senior fraud prevention officer with FFAUK.
Banks will never
Source: Financial Ombudsman
Mrs Sadjady did, eventually, get her money back.
Santander admitted they should have notified the Halifax more quickly about a possibly fraudulent transaction, and refunded her £3,000.
The Halifax admitted it failed to act fast enough in contacting other banks which received the money.
Initially it returned £4,490 to her account, but after being contacted by the BBC, agreed to compensate her for the full amount.
Similarly the Metropolitan Police agreed to open an enquiry, having initially claimed there was no realistic prospect of a conviction.
After the BBC supplied the recordings, it said that detectives from Ealing CID would consider all the available evidence.
No doubt they will also be studied to see how banks' security procedures can be tightened up.
After all, this fraud was carried out by men who started out with nothing more than someone's name, their telephone number, and the knowledge they banked with Santander.
The Moneybox episode that includes this story will be broadcast on Saturday, 5 September at 12:04 BST.
Three men who tricked their victim into handing over her £12,000 savings on the phone have been caught on tape.