Mr Lindsay is to retire at the end of 2012, after holding the top job for 13 years.
He recently oversaw the opening of ASB's first branch outside North Lanarkshire following a cash injection from a group of Scottish business figures.
Mr Ashley will take up the post in January. He has run Scotwest, one of the largest credit unions in the UK, since 1996.
Jim Armstrong, president of the board of trustees of ASB, said: "Appointing Rod is a real coup for the bank as he has a proven track record and brings with him a wealth of experience.
"He is a perfect fit for ASB having been at the helm of a member-based mutual organisation with an ethos very similar to ours."
Mr Ashley said: "I am looking forward to the prospect of a new challenge."
Mr Lindsay, who will still act for ASB in a consultancy role, said: "I'm proud to have been part of the success of ASB, and I look forward to retiring after almost 40 years in the industry."
Last year ASB opened a branch in Falkirk in the first stage of a £10 million expansion plan backed by Stagecoach founder Sir Brian Souter and Kwik-Fit founder Sir Tom Farmer.