The banking group at the centre of the NatWest IT meltdown raised hopes that it was finally getting to grips with the crisis on Wednesday.
RBS Group said the vast majority of NatWest and RBS accounts had now been free from disruption for two days as it works to clear up the chaos left by its IT failure last Tuesday night.
However it is still having greater difficulty restoring problems for its Ulster Bank customers, but a solution applied to the other two banks was successfully applied to Ulster Bank overnight.
It also said it is making progress in whittling down the numbers of remaining customers who are still experiencing problems.
The beleaguered banking group has scrapped its corporate hospitality at Wimbledon as it grapples with problems which have left some customers in limbo for more than a week, with some unable to access their wages and others facing delays moving home or disrupted holidays.
RBS Group said in a statement today: "Account balances have registered as normal for the vast majority of RBS and NatWest customers over the last two days.
"We can also confirm that the small amount of outstanding unprocessed transactions continues to reduce, and significant further progress is expected today.
"As previously cautioned, however, progress towards a completely normal service is likely to be affected by the significant stress on the system over recent days."
The group said that while Ulster Bank customers are likely to continue facing "significant problems" for the rest of this week, the success of last night's IT solution had increased its confidence that it will restore a full service by Monday.
It added: "We apologise once again for the unacceptable service customers have experienced and thank them for their patience."
Bank branches have extended their hours this week and the number of call centre staff has been doubled to clear up the problems.
Speculation has mounted over the cause of the problems, with unconfirmed reports suggesting that an inexperienced technician in India was behind the computer meltdown.
The problems reportedly began after a swathe of information was erased during a routine software upgrade for the banks, affecting the back-up systems as well as the live computer.
The deleted information is said to have been painstakingly re-entered into RBS Group's computer system.
A statement from RBS Group did not confirm whether the problems began in India but simply said staff in Edinburgh are working to put the issues right.
The statement said: "We have been clear we will fully investigate the causes of this incident. But we hope people will understand that right now our complete focus is on fixing this problem and helping our customers.
"The management and execution of the batch process is based in Edinburgh at the Fettes Row Data Centre, as is all of the current work to resolve the problem."
Frustrated customers said their patience was wearing thin, despite the progress the banking group said it has made.
One wrote on NatWest's website today: "Since last Tuesday not been able to access (my) account balance at all from my local cashpoint.
"Keeps telling me: 'Contact your bank'.
"Not easy when every other customer is trying to do the same! Card and service useless, time without access unacceptable!"
Another said they had endured "eight days of hassle" and the money they were expecting had still not shown up.
The customer wrote: "I have kept calm for eight days but I'm getting rather sick of it now."
Comparison websites said they have not seen a surge in bank account searches as a result of the problems, but they have suggested that customers' decisions on whether or not to abandon the group could hinge on how well it handles complaints over the coming weeks.
RBS Group has promised that no one will be left "out of pocket" and urged people to keep records of their losses, but it is yet to give details on exactly how people will be repaid.
Banks have a duty to return people to the position they would have been in if the mishap had not occurred, but some customers have argued they should receive extra compensation for inconvenience on top of anything they are owed.
Bank of England governor Sir Mervyn King has said the bank should be subject to a "detailed investigation" over the affair.
Consumers have been urged to be vigilant to make sure their credit scores are not adversely affected, although the banking group has said it is working to make sure no one's credit rating is blotted.
Meanwhile, HM Revenue and Customs said: "We are sympathetic to any customer experiencing genuine difficulties in not being able to pay their tax bill."
It also advised that those affected by the issues should seek help from their bank in the first instance.