Tesco Bank's chief executive has blamed Benny Higgins admits systematic failure for the money taken from 20,000 of its customer accounts.
He said the bank knew "exactly" what the attack was, but could not say more because it was part of a criminal investigation.
He said all affected customers would be refunded by the end of Tuesday.
About 40,000 accounts saw suspicious transactions over the weekend, of which half had money taken.
The bank is still blocking current account customers from making online payments using their debit card, a temporary measure it put in place on Sunday.
But it said customers would still be able to use their cards for cash withdrawals and chip and pin payments, while bill payments and direct debits would continue as normal.
"We are working hard to resume normal service on current accounts as soon as possible," said Mr Higgins.
Earlier, the bank confirmed some accounts "have been subject to online criminal activity, in some cases resulting in money being withdrawn fraudulently".
Mr Higgins also apologised for the "worry and inconvenience" that customers have faced.
One cybersecurity expert said this could be an unprecedented breach at a British bank.
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