Barclays said Agius, who has chaired the bank for six years, would remain in his post until a successor was found.
“I am truly sorry that our customers, clients, employees and shareholders have been let down,” Agius said in a company statement, less than a week after the bank was fined over the rate rigging scandal.
Barclays on Monday added that it would launch an independent audit that would “undertake a root and branch review of all of the past practices that have been revealed as flawed since the credit crisis started.”
Barclays insisted that it would establish “a zero tolerance policy for any actions that harm the reputation of the bank.”
Britain’s Business Secretary Vince Cable on Sunday backed calls for a criminal investigation into bankers involved in the scandal, which cost Barclays £290 million ($450 million) in fines.
The true cost of the fallout has been much bigger however, with the bank’s market value losing billions of pounds last week following a share price collapse.
Traders at the bank are suspected of manipulating the Libor rate, which plays a major role in international financial markets and affects businesses and consumers, in order to skew the markets in their favour.
“Barclays has remained resilient throughout the crisis, and has worked hard to ensure that today it is a strong, well capitalised and profitable business,” Agius said on Monday.
“But last week’s events — evidencing as they do unacceptable standards of behaviour within the bank — have dealt a devastating blow to Barclays’ reputation.
“As chairman, I am the ultimate guardian of the bank’s reputation. Accordingly, the buck stops with me and I must acknowledge responsibility by standing aside,” he added in the statement.
It emerged on Sunday that bailed-out The Royal Bank of Scotland (RBS) had sacked four traders over their alleged involvement in the affair, raising suspicions that the practice was widespread.
Prime Minister David Cameron on Friday reiterated his intention to bring Barclays chief executive Bob Diamond and others at the bank to account.
The bank’s actions distorted one of the main indicators of its financial health. Diamond was in charge of Barclays’ investment arm at the time of the suspected manipulation.
British lawmakers were due to question Diamond about the affair on Wednesday.
The pressure on Barclays has risen after British and US authorities last week fined the bank amid international probes into several top banks over alleged rigging of inter-bank rates.
Barclays is the first major financial institution to settle with regulators following investigations on both sides of the Atlantic.
Bank of England head Mervyn King on Friday said Britain’s banks needed a “real change in culture” following a series of scandals that rocked the country’s financial sector last week.
British regulator the Financial Services Authority on Friday said it had reached agreement with Barclays, HSBC, Lloyds and RBS to compensate clients for mis-selling interest rate hedging products.
Also last week, a severe IT meltdown at RBS left millions of customers unable to complete transactions.
RBS chief executive Stephen Hester and Diamond have both said they would forgo their bonuses for 2012 owing to the recent scandals.