The operation, to be called the New Foundation Bank, is modelled on Sainsbury's Bank, the joint venture between Bank of Scotland and the supermarket group that pioneered branchless banking in the UK in 1997.
Bank of Scotland will be the majority shareholder in the new venture and provide the know-how, while sub-contracting the operation of the call centre and back office to Marshall & Illsley, a US financial services group based in Wisconsin.
Applications for regulatory approval were filed with the American authorities yesterday.
No final decision had yet been taken on where and when to launch the venture. However, it is expected to be ready for launch within five months, starting probably in the Mid-west, where Dr Robertson has a strong following, before rolling out nationally as demand dictates.
Peter Burt, Bank of Scotland group chief executive, said the venture, if successful, would cost the bank some "tens of millions of dollars" in the first year. "There are no front-end costs. If successful, we will require capital to fund the bank. But it is structured in such a way as to limit the downside."
As with Sainsbury's Bank, the venture will start by offering high-interest deposits before expanding into other products.
Dr Robertson is a controversial figure in the US. A darling of the Christian right, he ran unsuccessfully for president in 1988 on an unashamedly right- wing programme.
He is best known for having founded the Christian Broadcasting Network, a religious television channel that he sold to Rupert Murdoch for $1.5bn in 1997.
Mr Burt said the idea for the venture had come from Bill Hendry, who heads up Bank of Scotland's existing US operations and who first raised the idea with the American TV evangelist a year ago.
Mr Burt said the bank would be managed at arm's length from Dr Robertson's other business activities, adding that it would be improper to mix what is a purely commercial venture with Dr Robertson's "charitable and religious works".
"He has a tremendous track record as a successful entrepreneur," Mr Burt said.
Dr Robertson, whose forebears left Scotland in 1695 - the same year Bank of Scotland was founded by a decree of the Scottish Parliament - said yesterday: "The objective is to provide the American consumer with a bank that is committed to service and value. Bank of Scotland brings unparalleled experience of service-oriented banking combined with a real understanding of the power of branding and marketing in start-up banking operations."