The airline said the incident has been resolved and all systems are working normally.
It's reported that approximately 380,000 customers had their private and financial details stolen from the British Airways website following a data breach.
The airline also made it clear that the stolen data did not include passport details.
Cruz said the British Airways informed customers affected by the attack on Thursday and advised them to contact their bank or credit card provider and follow their recommended advice.
"We take the protection of your personal information very seriously".
The airline was in the process of notifying affected customers and Alex Cruz, BA's chairman and chief executive, said it was "deeply sorry for the disruption that this criminal activity has caused".
"There was a very sophisticated, malicious criminal attack on our website".
BA is the latest major United Kingdom company to report such an attack - seemingly the largest since the owner of Currys PC World, Dixons Carphone, admitted in early summer that nine million of its customers had been hit by a data breach.More news: China's richest man Jack Ma will retire on Monday
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Will I have to get a new card?
In the U.S., Delta Airlines said in April that payment-card information for several hundred thousand customers could have been exposed by a malware breach months earlier.
A spokesman for the Information Commissioner's Office said they would be making inquiries about the data theft.
It was unclear how many people were affected.
Consumer advice website MoneySavingExpert says customers should monitor bank and credit card statements closely for signs of possible fraudulent activity.
Customers due to travel can check in online as normal.
British Airways experienced an IT-related crisis in May previous year when roughly 75,000 passengers were stranded after the airline cancelled more than 700 flights over three days because of system problems.