Mr Jenkins currently leads the Barclays Retail and Business Banking division and has been a member of the Group Executive Committee since 2009.
Marcus Aguis, Barclays chairman, said Mr Jenkins' appointment was based on his knowledge of the firm, his excellent track record over the past three years and his future vision for Barclays.
Mr Diamond resigned from Barclays in July after the firm was fined for misconduct relating to the LIBOR rate. Chief operating officer Jerry Del Missier also resigned.
Mr Aguis said: "Antony's appointment has the support of all the directors; we are confident that, supported by the Board and the executive committee, he will work quickly to take the Group forward.

"With the appointment of Sir David Walker to succeed me as chairman in November, and of Antony as chief executive today, I am pleased that the new leadership of the bank is settled."
Mr Jenkins said: "I am very proud to have been asked to lead Barclays, where I began my career nearly 30 years ago. Barclays is a strong universal bank, with many assets, including market leading businesses; talented and engaged staff; and long-standing clients and customers. But we have made serious mistakes in recent years and clearly failed to keep pace with our stakeholders' expectations.
"We have an obligation to all of those stakeholders – customers, clients, shareholders, colleagues and broader society – and a unique opportunity to restore Barclays reputation by making it the "go to" bank in all of our chosen markets. That journey will take time, we have much to do, and I look forward to getting started immediately."