Northern Rock has said it is "business as usual" as it waits to see whether a government promise to protect savers' deposits will reassure customers.
Shares in the bank rose 10% early on Tuesday after Chancellor Alistair Darling pledged the Bank of England would underwrite all existing deposits.
The bank hopes the move will slow the deluge of savers closing accounts.
It has taken out newspaper adverts to emphasise the company is "well-managed" and will not let its customers down.
Share in other banks also rebounded as trading on the stock market got under way. Alliance & Leicester, which saw its shares fall a third on Monday, rose 20%, while Bradford & Bingley climbed 9%.
The Bank of England also announced that it would inject £4.4bn into financial markets, offering cheap credit through a so-called emergency tender at a rate of 5.75% at the tender.
'Troubled times'
The full-page newspaper advert taken out by Northern Rock contains a statement signed by the bank's chief executive Adam Applegarth designed to reassure savers.
"The simple fact is that the chancellor has made it clear that all existing deposits in Northern Rock are fully backed by the Bank of England and are totally secure during the current instability in the financial markets," the statement says.
"These have been troubled times but Northern Rock will prevail."
The move comes after three days of long queues at branches up and down the country as customers withdrew more than £2bn of their money - about 8% of Northern Rock's total deposits.
There were still queues of about 70 people at a Northern Rock branch in Golders Green on Tuesday morning waiting for the bank to open even after the government's unprecedented move to restore confidence in the British banking system.
But in its home town of Newcastle, the streets around the bank were quiet.
'Solvent business'
Banks are already covered by the Financial Services Compensation Scheme which protects 100% of the first £2,000 in any bank account and 90% of the next £33,000 - giving a maximum payout of £31,700 if a bank did go bust.
But under the measures unveiled by Mr Darling, Northern Rock savers would not lose a penny, regardless of how much they had deposited.
In effect, this would nationalise the Northern Rock if it went out of business, potentially leaving tax-payers to pick up the bill.
This has been criticised by the Conservative Party, but Mr Darling, the Treasury and the FSA have repeatedly said that Northern Rock is a solvent business.
FSA chairman Callum McCarthy told the BBC that he welcomed the move by the chancellor.
"The purpose of this is not to save the Northern Rock per se," he said.
"It's to make sure that there's not a negative effect on the banking system overall."
Mr Darling said that people could continue to take money out of the bank, but that if they choose to leave it there it would be secure.
BBC economics editor Evan Davis said the chancellor's offer was a "dramatic step" and was, essentially a "money back guarantee".
"Why have they done it? It's because panics can get out of hand," he said.
"And even if a panic is irrational, if everyone else is rushing to get their money, it is quite rational to want to get your money before it runs out."
Northern Rock has said that it has not used any of the Bank of England's facility for funds offered to it.
Northern Rock applied to the Bank for emergency funding last week.
It has struggled to raise money to finance its lending ever since money markets seized up over the summer.
Unlike most banks, which get their money from customers making deposits into savings accounts, Northern Rock is built around its mortgage business.
It raises most of the money which it provides for mortgages by borrowing from banks and other financial institutions.