The plans follow the announcement in May that Sainsbury's has taken full control of the bank, buying Lloyds Banking Group's 50% stake for £260 million, in a move chairman David Tyler said would "allow the full potential of the bank to be realised".
Sainsbury's Bank employs more than 350 people at South Gyle in Edinburgh, it achieved a 20% increase last year, and it is recruiting for a similar increase by the end of 2013.
Chief executive Peter Griffiths said: "We have been delivering strong, sustainable growth over the past few years, and as we enter a new phase in our history, it is appropriate we have new offices that can support our growth plans.
"We have a loyal and talented workforce and remain committed to keeping our office in Edinburgh where the employment market for financial services is very strong."
The bank will take over the whole of the 83,000sq ft building in Edinburgh Park vacated by Aegon as part of its downsizing last year.
The 15-year-old Lochside Park offices, opposite Aegon's headquarters and formerly housing its asset management and other operations, are now being refurbished as part of the deal with landlord Aviva Investors.
Angela Lowe, partner at Sainsbury's agents Cushman & Wakefield, said: "This is a significant letting and great news for Edinburgh."
Agents Jones Lang Lasalle said it was the biggest office let in Edinburgh since Edinburgh council took Waverley Court as its new headquarters in 2004, and the biggest private sector let in a decade, though only narrowly topping Blackrock's let of a new 80,000sq ft headquarters in the capital announced last year.
The move will see all Edinburgh-based Sainsbury's Bank staff transfer from South Gyle in late 2014 to Lochside Avenue. Julian Cobourne at Aviva Investors said it was "arguably the best specified building on Edinburgh Park" and had justified the investor's strategy to hold and refresh.
Among Edinburgh Park's tenants are JP Morgan, Diageo, HSBC and BT, and Jardine Lloyd Thompson has recently taken space there.
Cameron Stott at agents Jones Lang LaSalle said: "The opening of the tram stop will add to staff accessibility and is clearly already having a major impact on occupier decisions. Considered together, these factors are resulting in a significant increase in the level of occupier activity in West Edinburgh this year."
He added: "It is a general reflection hopefully of the improving economic circumstances where some companies are looking to expand, and also because occupational costs are significantly cheaper than they were 10 years ago."
While rents at Edinburgh city centre's latest prime office development Atria are touching £30 a sq ft, those at Edinburgh Park have dropped from £25 to around £15.
Owen Kelly, chief executive at Scottish Financial Enterprise, commented: "It is very welcome and a real vote of confidence in Edinburgh."
He said it added to the recent investments of Tesco Bank and Virgin Money in the banking sector, and recognised Edinburgh's combination of financial service history, exceptional skills base, "and value for money".
The bank grew its customer accounts by 8% last year to 1.5 million.
Tesco Bank employs 2000 in Glasgow and Edinburgh, though its profit fell by 15.1% in the first half-year as it was hit by tough competition in insurance, falling legacy commission payments, and charges for mis-selling payment protection insurance. Virgin Money recruited 200 for its Edinburgh headquarters in 2010 and has promised further expansion.