Brown, who worked in a number of senior roles at RBS for 12 years until 2017, will become chief executive of Sainsbury’s Bank on June 19.
He replaces Peter Griffiths, who has run the business since 2012.
Brown’s most recent role at RBS was head of the bank’s Williams and Glyn brand, which it had planned to spin off as it was one of the conditions the EU set in return for approving the government's 2008 £45.5bn bail-out.
However, RBS avoided the forced sale of Williams and Glyn after agreeing to set up a fund to support challenger banks.
Other roles Brown held at RBS include chief executive of the lender’s Ulster Bank and head of retail and commercial markets in Asia.
"Jim has a wealth of experience and a strong track record of leading banks through significant change, which will help him lead Sainsbury's Bank through the next stage of its journey,” said Sainsbury’s chief executive Mike Coupe.
In the year to March 9, underlying operating profit at Sainsbury’s Bank fell to £31mln from £69mln the previous year due to bad debt charges following the adoption of new accounting standards, and a more cautious approach to unsecured lending.
Sainsbury's said on Tuesday it was focused on improving its bank's digital offering to keep and attract customers amid tough competition from online rivals.
In April, Sainsbury’s abandoned a plan to merge with Asda after the UK Competition and Markets Authority rejected the deal.