Midland Bank is promising to get new customers' first cheque book and cheque card to them within five working days of transferring their accounts.
It also pledging to get their Pin number to them within seven working days, and to authorise all standing orders and direct debits at the same time as the account is opened. If the service guarantee is not met, the bank will pay pounds 10 for each transgression.
The offer is being made in a drive for new business. Keith Whitson, Midland Bank's chief executive, said: 'We are looking to attract more than 10,000 new accounts.'
Recent research found more than 8 million people were dissatisfied with their bank accounts and wanted to transfer. But because of the problems associated with switching, only 150,000 customers a year move from each of the major banks.
Mr Whitson said: 'If we get it wrong we will do our utmost to correct the matter expeditiously . . . If the cheque book takes six days to arrive and we fail to pay three direct debits then we will put things right immediately and give the customer pounds 40.'
Midland said that until recently the number of new accounts it was taking on was equivalent to the number of people who were transferring, but claimed that recent new initiatives meant that it was adding customers to the 5 million it already had.
Barclays, National Westminster and Lloyds said they had no immediate plans to follow Midland with a guarantee backed up by a cash payment, but would negotiate compensation individually.
They said that on the whole they transferred customers within the same time as that guaranteed by Midland. They aim to get new cheque books and cards to new customers within five working days of opening the account and to get Pin numbers to customers within a few days of customers receiving the card.