The board of the Dundee-based firm announced it had agreed the terms for a takeover by America's Cardtronics, which will value i-design at £8.5m.
News of the deal sent shares in the Aim-listed firm surging and it closed up 157%, 36p at 59p.
The firm said the 60p per share cash offer is at a 51.1% premium to the average closing price of around 39.7p per share during the 12 months prior to yesterday.
Ms Stewart, who founded the business in 1991, has a 16.1% stake and is set to be the biggest beneficiary of the proposed deal.
Jim Faulds, the former chairman of Dunfermline Building Society who chairs i-design, is in line to net more than £650,000 for his 7.8% holding in the firm.
Texas-based Cardtronics, the world's largest non-bank operator of cash machines, said the acquisition would enhance the range of services it offers.
i-design supplies software that lets banks use ATMs for marketing and to run adverts for third parties. In a recent campaign, HMRC used i-design's technology to remind cash machine users about filing tax returns.
Cardtronics said the takeover will allow i-design to win more business from an enlarged customer base.
i-design said Cardtronics expects the Scottish firm's existing management and employees to play a key role in the continuing growth of the business.
Cardtronics indicated the offer will provide continuity of employment for staff, said i-design. The firm employs 30 people in Dundee and five in London. i-design added: "Cardtronics does not intend to change the location of the i-design Group's places of business [Dundee or London]."
Ms Stewart, 43, will be managing director of i-design and become a member of the Cardtronics executive committee, on a base salary of £120,000. i-design directors said the deal would provide the opportunity to achieve faster growth than would be possible in its current form, amid challenging conditions.
i-design has had to grapple with the fallout from the banking crisis and the economic downturn. "The i-design directors believe that the advertising market will remain very challenging in the short term," the company said.
Ms Stewart told The Herald: "The board is very pleased with the outcome and the value they have seen in us.
"The outcome is good for shareholders and for staff as a whole. Cardtronics is going to be supporting us all the way in terms of keeping the Dundee office and London and supporting our growth through their channels."
The deal provides a vote of confidence in the business which Ms Stewart started after doing a summer job at the former Dundee manufacturing plant of NCR, the US cash register giant,
She founded the business as a multimedia design agency working for banks and diversified into providing sophisticated functional software.
The company listed on Aim in July 2007. It had a market capitalisation of £9.5m, after raising £3.5m at 67p per share.
Northland Capital Partners told clients the deal looked like "a good result for shareholders that kept the faith with i-design".
Cardtronics has received irrevocable undertakings to accept the offer from shareholders in respect of around 62.2% of the share capital, including 24.6% held by board members.