TBWA\London, the incumbent, was the only other agency invited to pitch for the business, worth £10.2 million.
The review was overseen by The One Account's managing director, Jayne-Anne Ghadia. The appointment reunites the CHI founders Johnny Hornby and Simon Clemmow with the account on which they worked when they were at TBWA. CHI is already a RBS roster agency having won the brief to launch mc2, the company's new credit card.
A review of the business has been expected since TBWA won the Abbey National account in April.
Hornby said: "Our brief was to find a big idea for The One Account and we are excited about the one we have come up with. It is a long-term idea that really defines what the account is about."
The cult 70s children's cartoon series Hector's House was revived in January by TBWA to relaunch The One Account, formerly known as Virgin One.
RBS acquired The One Account from Virgin and AMP in July 2001. It was forced to rebrand the business after the licence to use the Virgin name expired at the end of last year.
It has not yet been decided what the future holds for Hector, but Hornby admitted his days were numbered.
At the time of its launch, Virgin One was the original all-in-one account that worked by holding customers' mortgage and savings in one place.
It now competes with the likes of The Woolwich and First Direct in a fast-developing market.