Mr Pym has held a number of high-profile roles in the UK over the past few years.
He was drafted in as chairman of the troubled Co-Op bank when serious governance issues were exposed.
The bank announced last April that it made a £2.5bn (€3.1bn) loss for 2013 — the worst performance in its 150-year history.
He is also chairman of UK Asset Resolution, which was the bad bank set up to wind down the toxic assets of Northern Rock and Bradford & Bingley.
Mr Pym will continue in this role. He was formerly chief executive of the Alliance & Leicester building society between 2002 and 2007.
“I am delighted to accept the invitation to be the next chairman of AIB and I look forward to contributing to the progress that has already been made in rebuilding a strong bank,” he said.
“Irish taxpayers have made considerable sacrifices in supporting the banking system and I am very conscious of my responsibilities in that regard.”
Mr Hodgkinson joined AIB in October 2010. The bank is 99.8% state-owned following a €21bn injection of taxpayer funds.
At the bank’s AGM in June, Mr Hodgkinson said he was confident that as it returned to the private fold, it would be able to pay back the Government’s entire investment in the bank.
It is expected that the partial privatisation of AIB will begin in the first quarter of 2015.
AIB returned to profitability earlier this year, according to group chief executive, David Duffy.
It will release its results for the first six months of the year on July 30.
AIB is currently undergoing the ECB’s comprehensive assessment of the banking system.
This involves an asset-quality review and stress tests.
The results of these are due to be released in November and will determine whether the bank is sufficiently capitalised to withstand future losses.
Finance Minister Michael Noonan said in a statement: “I am pleased to welcome Mr Richard Pym to his new role at what is an important time for the bank as it continues its transitions from a period of restructuring into one of recovery.
“Mr Pym has a wealth of experience in the UK banking sector and I am confident that he we will prove to be a strong chairman of AIB Group plc.
“I look forward to working with Mr Pym to ensure AIB continues to make progress so that it can support the recovery taking place in the Irish economy and begin to return capital to the State.”